ISQS 5349 (Dr. Westfall)

47497: PROC SYSLIN and SUR

Anonymous — Sun, 26 Apr 2009 22:57:28 GMT

The Freedman article distributed in class, wikipedia, and class discussions (4/23 @ 1:08:14) have used the economic theories of supply and demand as examples of endogeneity and/or reciprocal causation. Both state that a variable is considered endogenous when it can be predicted by another variable in the model; more specifically, the equilibrium price of a good or service is endogenous because suppliers change their price of a product in response to demand and consumers change their demand for a product in response to changes in price.

In a recently read article, the authors examine the substitutability of ATMs and EFTPOS (electronic fund transfers at point of sale, i.e. debt cards) by consumers when purchasing a good. The authors argue that the consumer’s choice of payment method (ATM or EFTPOS) is associated with (1) the ATM transaction fee, (2) the size of the ATM network, as measured by the number of ATM machines in a particular bank’s network, (3) ATM transactions by members, (4) ATM transactions by non-members, and (5) EFTPOS transactions.

There are 46 banks used in this study and data is collected from 1997 to 2003. The empirical methodology of this article involved using a system of five equations estimated using a three stages least squares (3SLS) with fixed effects.

First, is a three stage least squares regression versus an OLS used because of endogeneity and reciprocal causation among all five variables? For example, there is endogenity in (i) the transaction fee a bank charges for the use of an ATM machine in its network and (ii) the size of the ATM network because banks determine the fee they will charge for ATM usage based on the consumer demand, and consumers’ demand for ATMs will be affected by the transaction fee charged for use. Likewise, a bank’s choice to expand its network will be a function of the fee it can charge for usage and consumer’s demand for ATMs in a network will be affected by the transaction fee charged by the network? As well as because the data is panel data with autocorrelation and cross-sectional correlation since the data is for the same banks (within subject) across time (4/9 @ 40:15 and 4/23 @ 41:26).

However, would not a maximum likelihood model be “better;” possibly using the PROC SYSLIN and SUR procedures to analysis the five simultaneous equations, one for each dependent variable?

Furthermore, would not a random effects model be preferable to a fixed effects model? There are 46 banks or levels in the data and this is relatively large (more than 20, but less than but less than 133 that were in the “major” example we analyzed on 4/14 @ 52:47) and we are not concerned with each bank, but more with the cross-sectional “average” bank (4/14 @ 1:13:27).

46939 – Correlated error model for the multilevel data

Anonymous — Sun, 26 Apr 2009 05:13:30 GMT

In the 4/9 class (at the 0:31:42 mark), we were discussing the correlated error model in the multilevel data. The model is good to applied on the hierarchy model, e.g., the firms that share the industrial commonality are correlated, and we have to account the correlation.

One of my research topics is to evaluate the team performance on the improvement projects. To evaluate the team performance, I need to identify the important factors which are related to team performance on performing the improvement projects. A survey was conducted to collect the required data. The data set includes 51 teams from 6 organizations, and each team has 4 to 13 members. The data set is multilevel data with 3 levels: organization level, team level, and member level. Therefore, I think a correlated error model with the multilevel analysis is good to applied on the data. The dependent variable is the percentage of goal achievement (Y) in the team level. The potential predictor variables are the organization (ORG), goal difficulty (GDF), and project experience for each team member (TMEXP). The class variable is organization (ORG). The dependent variable goal difficulty is in the team level and the project experience for each team member (TMEXP) is in the member level. To consider the effect of organizations on the percentage of goal achievement (Y), I start with considering only the data in the organization level and team level. The model can be formulated as follows:

Y_ij = (β₀ +ϒ_0i) + (β₁+ ϒ_1i) GDF_ij + ε_ij

where ε_ij ~ N(0, σ²)

In this model,

i is the index of organizations. i = 1, 2, 3, 4, 5, and 6;

j is the index of teams in organization i. So, j = 1, 2, …, n_i ;

β₀ is the organization level intercept as the overall mean;

ϒ_0i is the random deviations from overall mean;

β₁ is the fixed effect from the GDF variable;

ϒ_1i is the random effect from the GDF variable;

ε_ij is the random error associated with the j^th team in the i^th organization;

The experience of each team member (TMEXP) is another potential factor that I am interested. However, the study focus on the team level performance and TMEXP is a variable in the member level. How should I do to evaluate the effect of team member experience (TMEXP) on team performance? How do I formulate the variable TMEXP in the correlated error model with the multilevel data?

43161 - Interactoin model for validating my research hypotheses

Anonymous — Sun, 26 Apr 2009 05:05:43 GMT

Measuring e-Commerce success is a challenging task and very complex entities in the IS field. I am trying to develop a new conceptual model for measuring e-Commerce success and empirically test the proposed model. The proposed conceptual model extends DeLone and McLean's IS success model (2004) by adding new constructors for the context of e-Commerce. The objective of this research is to investigate website quality factors, their relative importance in selecting the most preferred website, and the relationship between website preference and financial performance.

I believe that the regression analysis could be applied to validate this model and seek some meaningful findings of this model.

To do so, the following two hypotheses will be tested:

H1: Four website quality factors, such as Information Quality (IQ), System Quality (SQ), Service Quality (SVQ), and Trust (T) will impact website preference (WF).

H2: Customers' Perceived Intention (CPI) like awareness, reputation, and price savings about the website will moderate the effects of four website quality factors on website preference (WF) as shown in below figure.

Four website quality factors ----------------------------------> Website preference

Customers' Perceived Intention

To test these hypotheses, the regression model would be:

WF = beta0 + beta1 (IQ) + beta2 (SQ) + beta3 (SVQ) + beta4 (T) + beta5 (CPI) + beta6 (IQ*CPI) + beta7 (SQ*CPI) + beta8 (SVQ*CPI) + beta9 (T*CPI) + epsilon.

In this class, to understand this interaction concept, we have covered many examples and homework questions relating with the interaction models, two-way ANOVA, or GLM ANCOVA with interaction. However, our examples have looked at not only interaction involving categorized groups or qualitative observations, but also interaction between two predictor variables.

Here are my questions.

1. Along with my research objectives and hypotheses, is this interaction model correct?
2. If there are some moderator constructs in the conceptual model or exist multiple interaction between variables, do we have to make different interaction model for each moderator and test it separately?
3. CPI will be measured by scores (1-100). In this case, even though measurement of CPI is not category or qualitative data, can above interaction model be switched to ANOVA model? (In fact, I don't think so.) Otherwise, if CPI is a rank, is it possible?

41214--Fixed effect and random effect and model selection

Anonymous — Sun, 26 Apr 2009 04:44:35 GMT

41214--Fixed effect and random effect and model selection

In the class, we discussed fixed effect and random effect and then mixed models. Dr. Westfall suggested that if the number of levels are large, say, larger than 20, the random effect model is more parsimonious with less parameters.

Let’s look at two models, one is a fixed effect model in a regression form, the second is a random effect model.

(1) Y_i=b₀+b₁I₁+b₂I₂+b₃I₃+…+b_j-1I_j-1+e_ij e_ij~_iid N(0, σ²)

i=1, 2, 3, …, n (number of observations)

I₁, I₂, … I_j-1= indicator of level 1, level 2, …, level j-1 (there are j levels, the last one is the left out category).

(2) Y_ij=m+a_i+e_ij e_ij~_iid N(0, σ²), a_i~_iid N(0, σ_a²), {a_i} ind {e_ij}.

i=1, 2, 3, …, n (number of observations)

j=1, 2, 3,…, j (number of levels)

In model (1), there are j+1 parameters: b_0,b_1,b_2,…b_j-1, σ.

In model (2), parameters are: m, σ, σ_a. I’m not sure whether a_i is a parameter or not. I think it is the random effect not a parameter.

Comparing the model (1) and (2), model (2) is much more parsimonious in terms of the number of parameters. However, model (2) seems “mysterious” to me. Given an observation (knowing it belongs to which level), and the estimates of coefficients of model (1), we can estimate the Y_hat. However, given an observation, and the estimates of parameters of model (2), we need a_i to get the estimated Y (Y_hat).

Questions:

1. Where does a_i come from?

If there are j levels, there will be j random effects, one for each level. It seems that a_i are parameters, there are j a, one for each level. Then it is not parsimonious any more.

If a_i are not parameters, can a_i be determined by m, σ, σ_a? We can have the best linear unbiased predictor (BLUP) for a_i. To get the BLUPs, do we need more parameters to be estimated? According to the book link on Dr. Westfall’s course webpage (http://books.google.com/books?id=d5vhFpRIqiAC&pg=PA254&lpg=PA254&dq="Best+Linear+Unbiased+Predictor"+BLUP&source=bl&ots=xee_sba24O&sig=MfCxaADGj2R9sSxtCTH2HJSX_r0&hl=en&ei=UTTJSevlDsLrnQelwr2UAw&sa=X&oi=book_result&resnum=4&ct=result), the BLUP for a_i can be expressed as [σ_a²/( σ_a²+v_j)](y_j_bar-m), where v_j=Var(y_j_bar)= σ_j² /ni,

y_j_bar is the mean of y at level j, and σ_j² is the variance of y at level j. Am I right, here? I don’t know whether y_j_bar is the true mean and σ_j²is the true variance, or they are what estimated from data.

In my current research project, there are many levels of y, and for each level, there is only one observation, therefore, there is no variance, v_j should equal to 0, the BLUP for a_i is [σ_a²/( σ_a²+v_j)](y_j_bar-m)= y_j_bar-m.

Therefor, y_hat= m + a_i = m +y_j_bar-m= y_j_bar. Because there is only one observation, y_j_bar is the observation value itself, I’m confused here.Here, each level (although only one observation) can be seen as produced randomly from a process, so random effect model can be used here, which reflect the concept of “model produces data”. Am I correct here?The above question may be too specific, but I really want to figure it out. The following questions are also related to my current research, it’s about model selection.In our class, we discussed model selection, the variance/bias tradeoff. One of the criterions for model selection is AIC, which uses likelihood and imposes a penalty on more parameters. As far as I understand, AIC and some other criterions mentioned in the class (adjusted R-squred, PRESS, etc.) are used to compare models which have the same dependent variable(s). however, in my research, there are two choices of the dependent variable, they are similar but different in degree. The two variables both fit the research question. How can I choose between them? Is there any way to evaluate them. Or only the field knowledge and the research question or focus would decide which one to use.

48998 Regression versus ANOVA

Anonymous — Sun, 26 Apr 2009 03:34:11 GMT

Dr. Westfall,

I have a potential study that I would like to explore, but I am not sure if ANOVA or regression is best to use. But I know you stated that ANOVA is a type of regression, and ANOVA is more realistic because it doesn’t assume that all data points fall on a straight line. However, in the context of a specific study, wouldn’t there be justification for using regression over ANOVA, or vice-versa? To better explain, I have setup the following study:

Nature: A professor wants to compare two pedagogical tools (e.g. computer simulation and paper case study) in a management class to determine which method provides more favorable” attitudes toward teaching method” among undergraduate students. The professor teaches two sections of the same management class.

Design and measurement: A 1x2 factorial design study will be used with one factor measured at two levels. The factor is “type of teaching method”: computer simulation and case study. The dependent variable is student’s “attitudes toward teaching method”, which is measured on a 5 point-Likert scale. Convenience sampling will be used to select participants. In the first class, students will be exposed to the computer simulation. Then the students in the second class will be exposed to paper case studies.

DATA: In this 1x2 ANOVA, we have a possible theta of interest: mean. I have selected this theta of interest because I want to determine if the mean of the class exposed to the computer simulation is larger, smaller or equal to the mean of the class exposed to the paper case study. As you can see below, m_cs is the true mean for the first class exposed to the computer simulation, and m_pc is the true mean for the second class exposed to the paper case study.

q = m_cs – m_pc

The statistical model is represented by the following:

Y_ijk ~ N (m_ij, s²) independent

But I want to control for “attitudes toward management classes,” which is also measured on a 5-point Likert scale, because this factor could be a confounding variable.

So this ANOVA model be written as the following: Y_ijk = m_ij + z_ij+e_ijk, where Z_ij is the “current attitudes toward a management class.”

For the same study, the regression model would be:

Y = b₀ + b₁ + b₂ + Z₁ +e, where Y = Attitude toward the teaching method, b₀ + b₁ is the mean for the first class, and then b₁ + b₂ is the mean of the second class minus the mean of the first class, and Z₁is the controlling factor. In regression, I can see whether using a computer simulation or a paper case study method would have a higher effect size. To my understanding, effect size translates into what a 1x2 ANOVA would provide (i.e. difference in means).

By comparing ANOVA versus regression, it appears that I am going to get similar answers. So if using ANOVA or regression is based on preference, would this serve as enough justification for using one of these methods? I feel that I am missing a key point on why ANOVA or regression would be used. I ask because when I conduct studies in my college, I always get directed to state that I am using ANOVA rather than regression. Thank you.

43292 ANOVA vs. Regression Model with Dummy Variable

Anonymous — Sun, 26 Apr 2009 03:24:10 GMT

ID: 43292

As I briefly describe my research, I am interested with the relationship between task structure (independent variable) and individual effectiveness (dependent variable).
In other words, this study considers

Y = individual effectiveness (we suppose this is from 1 to 10 high), and
X = perceived task structure of individuals, coded as 0 and 1, for simple task structure and complex task structure.

In one of research meetings, my professor told me the ANOVA model is more appropriate in order to analyze this relationship.
Based on above description, the ANOVA model is as follows:

Y_ij= μ_i+ε_ij, where E(ε_ij )=0, and where i = 0, 1 denotes task structure (simple or complex),
and where j = 1, …, n_i denotes number of individuals within group i.

I agree this opinion, since the ANOVA model can be used to compare two or more means to see if there are any significant differences among them.
However, according to my understanding, we can also use regression model with dummy variable:

Y_i= β_0+β_1 D_i+ε_i, where E(ε_i )=0, and where D_i = 0 if simple task structure, or 1 if complex task structure.

Then, these two models are always interchangeable? Or in my research, which is better for more realistic model?

In addition, my professor also told me the causal effect between X and Y cannot be inferred in the ANOVA model.
As such, I think it is hard to explain the causation in this relationship because the ANOVA model states the mean differences between two or more groups.
However, if we use the regression model, does the regression model can be more likely to predict causation (if we assume that hold every confounding variable fixed) than ANOVA model?
Otherwise, I am still wondering which model can explain the causal effect in this research?

43125 - Interpretation of Interaction Regression Models

hoh — Sun, 26 Apr 2009 02:31:14 GMT

We have learned about moderator variables and interaction models. Citing a research study published in Journal of Business Venturing (Zahra and Hayton, 2008), I want to ask about interpretation of interaction regression models with linear effects.

In this article, the researchers suggest that the effects of international venturing activities (independent variables) on financial performances (dependent variables) depend on companies' absorptive capacity (moderator). The international venturing was separated into six variables: related and unrelated international acquisitions, related and unrelated international alliances, and related and unrelated international corporate venture capital (CVC). All the items followed a five-point Likert-type scale (1=strongly disagree vs. 5=strongly agree). One of two measures to capture a company's performance in this study was the three-year average return on investment (ROE). The ROE variable is in its log-transformed state. The analyses controlled for several company-related variables and for country and industry effects-related variables.

One of the hypotheses is that "the strength of the relationship between international acquisitions and a firm's profitability (ROE) is positively related to its level of absorptive capacity" (p.200). In order to test the hypothesis, the researchers ran separate hierarchical regression analyses for ROE. First, Model 1 regressed ROE on the control variables. Next, Model 2 included the control and international venturing variables. However, the six international venturing measures were not significant whereas absorptive capacity was positively associated with ROE (p<0.05). The coefficient for absorptive capacity was 0.23 and the coefficients for the six international venturing measures were 0.09, 0.06, 0.08, 0.10, 0.05, and 0.09, respectively. The intercept was 0.73 in model 2. In the third step, the interaction term for related international acquisitions (i.e., related acquisitions multiplied by R&D spending) was added to the variables already in model 2. The analysis was significant (p<0.001), explaining 34% of the variance in ROE. The intercept was 0.59. Even though the coefficients for the six international venturing measures were 0.06, 0.06, 0.05, 0.10, 0.05, and 0.07, respectively, which were not significant, the interaction term (0.26) was also significant (p<0.01). Based on these results, there seems little doubt that international venturing activities may indirectly influence a company's ROE.

The information mentioned above allows us to develop two regression equations as follows:

Model 2: Y_predicted = 0.73 + 0.09X₁ + 0.06X₂ + 0.08X₃ + 0.10X₄ + 0.05X₅ + 0.09X₆ + 0.23X₇.

Model 3: Y_predicted = 0.59 + 0.06X₁ + 0.06X₂ + 0.05X₃ + 0.10X₄ + 0.05X₅ + 0.07 X₆ + 0.21X₇ + 0.26 X₁X₇.

where X₁=related international acquisitions, X₂=unrelated international acquisitions, X₃=related international alliances, X₄=unrelated international alliances, X₅=related international CVC, X₆=unrelated international CVC, X₇=absorptive capacity, and X₁X₇= related acquisitions multiplied by R&D spending.

The predicted Y in model 2 when X₇=0 is given by:

Y_predicted = 0.73 + 0.09X₁ + 0.06 X₂ + 0.08X₃ + 0.10X₄ + 0.05X₅ + 0.09X₆ + 0.23X₇=0.73 + 0.09X₁ + 0.06X₂ + 0.08X₃ + 0.10X₄ + 0.05X₅ + 0.09X₆.

Also, the predicted Y in model 3 when X₇=0 is given by:

Y_predicted = 0.59 + 0.06X₁ + 0.06X₂ + 0.05X₃ + 0.10X₄ + 0.05X₅ + 0.07X₆ + 0.21X₇ + 0.26 X₁X₇ = 0.59 + 0.06X₁ + 0.06X₂ + 0.05X₃ + 0.10X₄ + 0.05X₅ + 0.07X₆.

Thus, there is the difference between those predicted Ys. I wonder why the predicted Y in model 3 when X₇=0 differs from that in model 2? In model 3, it looks like that an increase in related international acquisitions has a larger effect on ROE when absorptive capacity is at a higher level than when it is at a lower level. It looks like that ROE changes by nine percent for a one unit increase in related international acquisitions while all other variable in the model 2 are held constant. However, given the interaction term, how many percents of ROE does change for a one unit increase in related international acquisitions while all other variable in the model 3 are held constant?

Reference: Zahra, S. A., & Hayton, J. C. (2008). The effect of international venturing on firm performance: The moderating influence of absorptive capacity. Journal of Business Venturing, 23, 195-220.

Final Question: Heteroscedasticity issues in Research

Anonymous — Sun, 26 Apr 2009 02:23:42 GMT

45297

The concept of heteroscedasticity is one that has been discussed throughout the entire semester. As a matter of fact homoscedastic errors are an important assumption of the classical regression model. The most central theme discussed in this regression class seems to be that the model produces the data and not the other way around. A model is meant to be a simplification of reality. All assumptions made for a particular model are about the reality that creates the data. Thus when an assumption is violated as in the case of the presence of heteroscedasticity, the classical model is no longer a reasonable representation of the reality one is trying to measure. A correction must be made for the heteroscedasticty in the data. The following are highlights of corrections for heteroscedasticity discussed in class.

Solutions for heteroscedasticity were given to be performing a log transformation, fitting the data with a beta distribution, quantile regression, robust regression and WLS which is a special case of GLS. In this case I am going to attempt to solve the heteroscedasticty problem in my data set by using GLS. GLS estimates are BLUE and have the correct standard errors and p-values.

In the data set I am going to ask my question about I analyze two regions: Western Europe and Eastern Europe. I want to test each region’s energy consumption as a function of its oil reserves. The regression function is as follows: Y_i= a₀+ a₁X_i+D_i+ ε_i.The error is normally distributed.

Where Di=1 for Western Europe and i=0 for Eastern Europe.

Yi=Energy Consumption

Xi=oil reserves

SAS performs a version of the Breusch-Pagan test using the SPEC option in PROC REG. By running the regression using my model Y_i= a₀+ a₁X_i+ D_i+ ε_i, it turns out that the Breusch -Pagan test statistic has a value of 12.78 which allows me to reject my null hypothesis of homoscedasticity. Thus it can be concluded that heteroscedasticity is present between Western Europe and Eastern Europe.

Question:

Philosophical:

As stated above it appears that there is heteroscedasticity present between Western Europe and Eastern Europe, thus there is between -subject variance, making the OLS estimator on the entire sample inefficient relative to the OLS estimator on each subsample. In this case, the subsamples, one for Western Europe and one for Eastern Europe, do not suffer from heteroscedasticity. Thus there is between subject variance but not within subject variance in my dataset. Is then the OLS estimate for the entire sample still inefficient relative to the OLS estimate each subsample? In other words, when there is variance between Western Europe and Eastern Europe, but not within the sample for Western Europe, and not within the sample for Eastern Europe, the overall OLS estimator will be inefficient relative to the OLS estimator of the subsamples. Is this correct?

Technical:

How would one perform Generalized Least Squares in this case? Is it correct to take the following steps: First I would fit Ordinary Least Squares to the model. Then I would take the squared residuals obtained from running this regression. Next I would regress the squared residuals on the explanatory variable and obtain the fitted values from this regression. Then I would insert these fitted values on the diagonal of the variance –covariance matrix of the epsilons. This will then result in a covariance matrix which is a consistent estimator. Finally I would use this covariance matrix in the final GLS estimation.

β-hat _GLS= (X’Ω^-1_hatX) ^-1 X’ Ω ^-1_hatY

Where Ω ^-1_hat is a consistent estimator of Ω.

Is this reasoning correct?

Generally one would like to weigh the equation by the standard deviation of epsilon, since this results in a variance that is no longer dependent on the individual observations. However this is not possible because the variance –covariance matrix of the true epsilons is not known in reality. Is that correct? If yes, how else can one improve the accuracy of the estimates?

I used SAS PROC MIXED to perform Generalized Least Squares. Is it correct that PROC MIXED in SAS performs its regression estimation based on Generalized Least Squares. And are the estimates Maximum Likelihood estimates under the normality assumption?

The problem with heteroscedasticity is that one cannot trust the t-statistics because the estimates of the standard errors are biased. The nice thing about White’s test is that the values will be correct whether or not heteroscedasticity is present. The ACOV option in SAS provides these robust standard errors. Is that correct?

"Snowball Sampling" & Correlation of Errors Concerns

Anonymous — Sun, 26 Apr 2009 02:13:18 GMT

My question this week relates to the potential problems encountered with using OLS when you have correlated error terms. I also tried to understand the various ways this correlation typically arises and how to model it to achieve better estimates and correct standard errors. I specifically reviewed our class material from 4/9/09 through 4/23/09 in designing this question.

I reviewed the several different examples given in class of correlated error terms. In within-subjects and repeated measures experiments, we learned that you should expect correlation between a subject and himself. In multi-level data we should expect correlation between subjects from the same level (such as company, major, etc). In time series and spatial data we should expect correlations between observations that are close in time and close in distance, respectively.

My question combines the correlation of errors concept with a sampling methodology that I learned about in one of my research methods classes. There is a technique for sampling called “Snowball Sampling.” Snowball sampling is used when you need to identify people for your study who are difficult to access. For example, in auditing research, many times researchers want to have working audit seniors complete their experiment or survey. However, unless you are able to get audit firms to allow you to give your study at an audit senior training session, it is very difficult to identify and gather those professionals to complete your study.

The idea in snowball sampling is to solve this problem by first identifying a small number of people that you know who fit your needs. For example, I could e-mail my experiment/survey to ten people that I personally know from working in public accounting. I would ask those ten people to also recommend others to me who fit my needs (i.e. other practicing audit seniors). The snowball begins to build. I would then contact those people for my study, asking them to participate and to recommend others who could complete the study. This process would continue until you have reached an adequate sample size. In this way, your sample is gathered in a “snowball effect."

In my methods class, we learned that the main problem in snowball sampling is “external validity.” Someone can challenge your ability to generalize from your results, because of the way you gathered your participants (i.e. not a random or representative sample). However, after learning in ISQS 5349 about the problems with correlated error terms and ways that correlations arise, I am wondering if correlation of errors is also a major problem with this sampling technique?

I am thinking that there very likely could be correlations between responses from a participant that I initially select and the participants that he recommends. Since the participant is recommending people that he knows and possibly is friends with, they likely share commonalities that might impact my study. Since I suspect correlations between people and the people they recommend, I would need to keep track of that information and possibly model it instead of just using OLS (depending on the severity of the correlation).

Based on the covariance structures we’ve examined in class and my a priori theories on correlation between people and people they recommend, I think that there are 4 good models up for consideration:

1) AR(1) – in this structure, the correlations get smaller as the “lag” increases. In my case, I believe that the strongest correlation would be between the person and the people he recommends. However, there could also be weaker correlations between the people the person recommended and the people they recommend (i.e. a lag of 2).

2) Toeplitz with Two Bands - this structure allows that there is a correlation between a person and the people he recommended (basically 1 lag), but no other correlations.

3) Spatial Power – This structure allows that the correlation depends on the “distance” between observations. In my example, the distance is smallest (i.e. most correlation) between a person and the people he recommended.

4) Unstructured - This structure is least parsimonious, but it does allow (realistically) that not every correlation is equal (i.e. all the “lag 1” correlations do not have to be the same, all the lag 2 correlations do not have to be the same, etc.)

Question – First, does my general logic and approach to modeling seem reasonable given this sampling technique?

I think if I just ran OLS without trying to model the potential correlation(s), I may get incorrect standard errors that might cause me to find significance that doesn’t really exist. Therefore, should I model for these potential correlations as I suggested above, checking for significance of error correlations by the p-value only? We have talked about how p-values don’t necessarily mean you have “practical significance.” How can I tell if I have “practically” significant correlation of errors? Is this just a matter of looking at parameter estimates under OLS compared to parameter estimates under GLS modeling for the 4 structures above?

49789: Quadratic Term

Anonymous — Sun, 26 Apr 2009 02:02:50 GMT

The phenomenon that I am interested in researching is a relationship between corporate social performance (CSP)and corporate financial performance (CFP). Several models have been proposed to explain this linkage, ranging from positive and negative to curvilinear model. For example, scholars argue that corporate philanthropy should be positively associated with corporate financial performance because corporate philanthropy facilitates stakeholder cooperation and helps secure access to critical resources controlled by those stakeholders. In contrast, other scholars take a negative stance, suggesting that corporate philanthropy diverts valuable corporate resources and tends to inhibit corporate financial performance. Recently, Wang, Choi, and Li (2008) theoretically integrated and extending these existing perspectives to propose a curvilinear model evoking the law of diminishing returns. Their estimation model is as follows:

corporate financial performance(t+1) = beta0 + beta1 * corporate financial performance(t) + beta2 * corporate giving(t) + beta3 * corporate giving(t)^2 + beta4 * dynamism(t) + beta5 * corporate giving(t) * dynamism(t) + beta6 * corporate giving(t)^2 * dynamism(t) + beta7 * control variables(t) + error terms (t)

To examine whether there is a curvilinear relationship between corporate giving and corporate financial performance, I focus on the coefficient of giving(beta2) and the other coefficient associated with giving (=beta3). The authors report that beta2 is 0.840 (p<0.05) and beta3 is -9.204(p<0.001). Interpreting these results, they explain as follows:

" the coefficients on both the linear giving term and the quadratic term were highly significant for both measures of financial performance (at least at the p<0.05 level). The positive coefficient on the linear term and the negative sign on the quadratic term are consistent with the predicted curvilinear(inverse U-shaped) effect of charitable giving on corporate financial performance. "

Their explanation that the relationship between corporate giving and corporate financial performance can be best captured by inverse U-shape makes sense to me. I can see that a significant quadratic term (p<0.05) indicates that there is statistically significant evidence of curvature.

To the contrary, our class discussion dated January 29, 2009 demonstrates a SAS example (toluca) testing for curvature using quadratic regression. The results is presented as follows:

19:38 Saturday, April 25, 2009 1 The REG Procedure Model: MODEL1 Dependent Variable: workhours Number of Observations Read 25 Number of Observations Used 25 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 2 252958 126479 51.30 <.0001 Error 22 54245 2465.67237 Corrected Total 24 307203 Root MSE 49.65554 R-Square 0.8234 Dependent Mean 312.28000 Adj R-Sq 0.8074 Coeff Var 15.90097 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 87.70964 58.61938 1.50 0.1488 lotsize 1 2.68204 1.86390 1.44 0.1642 lot_sq 1 0.00647 0.01333 0.49 0.6323

I can see that an insignificant quadratic term (p>0.05) indicates that there is not statistically significant evidence of curvature.

However, there is one thing that has not been clear to me. Am I concluding that there is evidence of curvature between corporate giving and financial performance because both linear and quadra terms are statistically significant? Or the statistical significant of quadratic term alone indicates curvature. Mathematically, it seems to me that what matters is quadratic term alone, as it determines convex (or concave), but I am yet to be sure if I am right. I would appreciate your clarification on this.

	Linear Term	Quadratic Term	Curvature
Corporate Giving-CFP	0.840(p<0.05)	-9.204(p<0.001)	Yes
Toluca	2.68204 (pr > t : 0.1642)	0.00647 (pr > t : 0.6323)	No

Variance/bias trade-off and multilevels models

Anonymous — Sun, 26 Apr 2009 01:37:03 GMT

45213

My question is related to the concept of variance/bias trade-off and multiple level models. I am particularly interested in the multiple level models. I would like to apply them to the study of entrepreneurial phenomena. My specific interest is on the double interaction between micro levels and macro levels. That is, the interaction between an actor (entrepreneur) and his/her context and vice versa. Sometimes these models are called “meso models”. There are two concepts that are interesting to study when examining “meso models.” The first is embeddedness. This concept explains that people are embedded in social networks (cultural factors), and these social networks influence people’s behavior (Granovetter 1985). The second concept is structuration theory, which says that there is a dual relationship between agency and structure. So, people cannot act without structure, and structure does not exist without people (Giddens 1979).Although highly realistic, these concepts have some constrains when trying to prove the theory empirically. For instance, case analysis is one the most used methods. However, “the case study approach that is largely based on qualitative evaluations still leaves unanswered the challenge of a broader and more formal empirical verification of these concepts [ a mixed embeddedness and immigrant entrepreneurship] and the arguments based on them” (Razin, 2002: p, 166 in Conclusion, the economic context embeddedness and immigrant entrepreneurs, International Journal of Entrepreneurial Behavior and Research, vol 8). In philosophical terms, the research that I am interested in can be categorized as constructivist. In this sense, this constructivist approach could have a drawback, which is the fact that it tends to be too descriptive and it does not allow for predictions pre-facto but for explanations post-facto. The argument is that understanding, explanation, or prediction is not sought in terms of factors external to the phenomenon. This is why qualitative analysis enables researchers to understand as much as possible one single phenomenon. Making the connection with concepts that we saw in class, a possible prediction variance of such models is almost infinitive simply because the probability that another similar phenomenon to happen is very close to zero. However, we are human, and as such, we share some communalities, which any researcher (positivist or constructivist) in social sciences try to uncover. Thus, how can researchers in a constructivist approach include more prediction capacity without losing descriptive capacity of such models? Are we “condemned” to face a variance/bias trade-off while trying to have prediction and description capacities? I am interested in understanding the concept of variance/bias trade-off in multiple level models. That is, how I can model the variance in a situation characterized by the interaction between different levels, without specifying an excessively complicated model. We saw in class that when I add more variables to a model, I add variability because in a model with more variables there are more differences between any prediction and the average prediction. In other words: more variables, more variance in the prediction and therefore less accuracy. In technical terms, my confusion starts when reading Singer’s paper about multilevel models. From this reading, I am thinking that there are different sources of variation that we need to account for, but I also think that I do not understand very well the differences in these sources of variation, especially when we talk about random effects and variance/bias trade-off. Let me briefly state how I understand Singer’s paper. Through the hierarchical models examples of students and schools, it seems that the idea is to add variables to reduce the variance created with the random effects. Then, when analyzing the results, if there is still variance to be explained, we need to decide whether another predictor should be incorporated. For example, she starts with the unconditional means model --with fewer variables, and then she includes a predictor at the school or level 2. She makes some calculations to show that the added predictor variable explains “the 69% of the explainable variation in school mean math achievement scores” (Singer 1998, p: 332). Thus, in this case the addition of a new variable reduces the variation in the dependent variable. So, at first sight this confuses me a bit because it might contradict the variance/bias trade-off. However, in a parenthesis Singer says “Note that this is not the same as a traditional R² statistic. This percentage only talks about the fraction of explainable variation that is explained. If the amount of variation is small, we might be explaining a large amount of very little!” (Singer 1998, p: 332-333. Singer references an article that discusses this aspect further, however we do not have access to the full text of this reference). In another comparison with the inclusion of a level 1 predictor (student level), she concludes that “Inclusion of student level SES has therefore explained (39.15-36.70)/39.15=6% of the explainable variation within schools. Comparatively speaking, then, school SES explains much more of the variation in school level math achievement than does student SES explain the within-school variation in student level achievement” (Singer 1998, p:336). She says that the word “explained” is crucial in our interpretation. When trying to make sense of the fact that adding more variables reduces the variance explained, I get to the conclusion that there are different sources of variance. Therefore, I think that I have different questions but all related to the concept of variance and perhaps with the more general issue of model selection. 1. In my research, I think it will be feasible to apply random effects. Especially in concepts of mixed embeddedness when not only cultural (embeddedness), but also economic, political, and legal factors can affect individual levels and vice-versa. Thus, random effects add the possibility to account for variation between several levels. Thus, the idea is to find variables that explain this variation. To the extent that the addition of these variables reduces the variance that we can explain, we decide to leave these variables or not in the final model. However, using the variance/bias trade-off concept, why will not the variables that we include (to account for the random effects) add more variance to the model? Or perhaps they are adding more variance, but the to predictive variance of the model…2. According to Singer, the word “explained” is crucial. Thus, do random effects refer to variance that we could explain adding variables, and variance/bias trade-off refers to prediction variance? Putting this in more simple words, the variance in random effects comes from the actual data that we see, and the variance in the variance/bias trade-off comes from possible data that we could see from the model? If this is correct, which variance refers to R²? According to the definition in my notes, it looks that R² is more related to the variance that we see from the data, but Singer’s parenthesis makes me think that this is not the case. Note: perhaps my question looks too long. This is not because I am trying to ask twice as many questions. But since I had to provide enough context so anybody can understand, I describe the main concepts and some key parts from the articles that I was thinking of.

48961-Understanding the contradiction between R-square and t-test in regression analysis

Anonymous — Sun, 26 Apr 2009 01:09:06 GMT

Dr Wetfall; In my reading I came across an unusual situation were the data from a million observations Xi to determine y produced a Fuzzball-like data graph. I was unable to post the graph, unfortunaley, but I got the regression model parameters and tests as a result of the regression analysis(Scratch) shown below from SAS/STAT Proc Reg.

Reg Procedure

Model 1

Dependent var: y

Number of observations 1000000

Analysis of variance

Source DF Sum of squares mean square F value Pr>F

Model 1 147 147 147.10 <0.0001

Error 999998 1001727 1.00173

   Root MSE                                                1.00086                               R-square                   0.0001

   Dependent Mean                             -0.00000420                          Adj R-sq                        0.0001

Coef Var                                              -23830403

Parameter Estimates

Variable            DF                 Parameter Estimate            Standard Error            t value       Pr>/t/

Intercept             1               -0.00002670                           0.00100                    -0.03           0.9787

x                          1                      0.01213                           0.00100                    12.13           <.0001

Initially, looking at the graph I would say there is no possiblity of trend and therefore Y does not depend on X. However, looking at the regression analysis results above, I note that the t value for the regression slope is highly significant but the R-Square statistic is .0001, meaning that X explains only .01% of the variance of Y. It seems like there is some condradiction in the results between the r-square value and the t-test. I do recall during this semester you mentioned several times that for a t-test on a hypothesis H0: Beta1=0 , with large sample sizes, even small values of Beta 1 will show statistical significance. I suppose that could be the case here, right? I do recall you talked about the power of the test increases with larger n. You mentioned that if the coefficient is statistically significant, we can conclude beta1 not = to 0 and not necessarily practical significance . I suppose then, in this case, the test is powerful and statistically significant. Now, looking at the t value of the slope(12.13), highly significant, could we conclude that X is important even though r-square suggests it is not? What could one write in a research paper about the conclusion of y depending on X in this case, I would qualify as unusual (contradictory scenario)?

49671 - Time series and error correlations

Anonymous — Sun, 26 Apr 2009 00:22:36 GMT

BACKGROUND INFORMATION

I am studying a Health Technology Assessment published by the Canadian Agency for Drugs and Technologies in Health. This particular issue looks at a new method for cervical cancer screening called Liquid Based Cytology (LBC).

The crux of the paper is comparing the new technology, LBC, with the current technology called Conventional Cytology (CC) or Pap-smear in terms of cost-effectiveness and clinical effectiveness. It also looks at the effect of triaging unspecified cases (cases that may or may not lead to cancer) using Human Papillomavirus (HPV) testing. The paper is a systematic review of many different studies comparing the two technologies, and makes suggestions of when the new technology is preferred. At the same time, it uses models to estimate how effective the technologies will be in the future.

The basic conclusion of the paper is that any screening method using HPV testing is preferred over no testing. However, it mentioned that more women in high-risk populations are getting the HPV vaccine and thus will change the interpretation of current models.

QUESTION

In class we have been discussing time series data, mixed models, error correlations and their applications. I have some questions of how these concepts can be applied in the example mentioned above.

The most significant factor preventing organizations from tracking women over time is the fact that there is no legislation in place that allows them to recall women at a particular time interval to track their health. So in practice, the methods for tracking of women in the population are opportunistic. However, there are women who do follow the prescribed screening of yearly tests. Am I correct in thinking that the model that would fit is a mixed effects model?

Furthermore, there is a time dependence issue and I am not sure how to address it. If high-risk populations are changing in terms of risk factors (more women have a vaccine), then the models to estimate the future effectiveness used in the original analysis will not be as accurate as they could be. The new models need to reflect that in ten years from now high-risk populations will look different due to the vaccine.

If we look at populations over time in order to see if and how cervical cancer incidence has changed and the role of screening methods, I think we need to use time dependent models. However, we are not tracking people as in a specific person i.e. Mrs. XYZ over time; we are tracking sets of populations i.e. women 15≥30 in a particular year and women 15≥30 three years later. There may be some overlap, meaning that there may be some people in the same cohort three years (or some other measure of time) later, but there will be some new people as well. So, how is this time dependence modeled? Would it be panel data since it does have a cross-sectional component? And can we treat the cohort of women as a “unit” even if the individual components change?

Finally, how does the time dependence affect the error correlation? I am aware that time series data has high error correlation that might decay as time passes. But, is this still true when the components of the “units” examined over time change perhaps even completely?

49671

49625 : How to model interaction between a fixed effect and a random effect

Anonymous — Sat, 25 Apr 2009 23:19:11 GMT

My question is about the mixed models, which are used when we consider some effects as fixed and some as random. I have learned in the previous courses that an effect being fixed or random has to do with the scope of inference, ie if we want to generalize the effect over and above the levels used in the study, then we consider the effect as random and if we are interested in some particular levels of factors, then consider the effect as fixed. Is it OK to follow this as a general rule? But if there is an interaction between the factor having a fixed effect and the factor having a random effect, would we model the interaction as fixed effect or random effect? I went through different documents to see how the interactions are generally modeled in designed experiments, which are very common in my field. In the SAS documentation for MIXED procedure, the codes for analyzing a split plot design (It is like a nested model, in which levels of one factor is applied to a larger plots called main plots and different levels of the second factor are applied in smaller plots within each main plot, called subplot) is given as follows.

proc mixed; class a b block; model y = a|b; random block a*block; run;

Where a and b are main plot and sub plot treatments (Level of fertilizer and variety of crop planted respectively). Here the interaction between the random block effect and fixed effect of fertilizer is modeled as a random effect. But you have also discussed in class that if interaction exist between two variables, then we can explain the effect of one variable only at the various levels of the other factor. Doesn’t it mean that the generalization argument about the random effect fails here? So how can we decide on whether to model the interaction as fixed or random effect?

45932 Appropriate Model for Housing Leverage

Anonymous — Sat, 25 Apr 2009 23:09:49 GMT

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I am currently interested in doing some research on predicting housing leverage, defined as mortgage debt outstanding divided by total value of all homes owned, among older households (those 55 years of age or older). After considering a few different statistical models presented in the course (discussed below), which statistical model would you recommend? Based on the ones I presented below, it appears that the ANCOVA model may be the most appropriate, but there is a more appropriate model?

Response Variable

Housing Leverage=Total Mortgages Outstanding/Total Homes Owned

Housing leverage, the response variable, is a ratio or percentage that is theoretically bounded from 0 to infinity. However, it is very unlikely that households can or will have leverage above 100%, with rapidly decreasing likelihood of being more leveraged as leverage increases beyond 100%.

Predictor Variables (chosen based on Life Cycle and Other Theories)

Predictor Variables	Type
Itemize Deductions?	Dummy
Have 10% Equity in Retirement Account?	Dummy
Age (Categories 55-64, 65+)	Dummy
Good Health	Dummy
Working	Dummy
Married	Dummy
Risk Averse	Dummy
Have Children	Dummy
Nonwhite	Dummy
Self-employed	Dummy
Adjusted Gross Income (AGI)	Continuous
Net Worth	Continuous

Ordinary Least Squares

Normality Assumption

Since our focus is older households (those 55 years of age or older), it seems that a theoretical distribution of housing leverage for this group could appear as an exponential distribution, where f(x)=5*exp^(-x/.2), thus saying older households are about 12 times as likely to have housing leverage of 0% than 50% and 150 times as likely to have housing leverage of 0% than 100%. While a beta distribution also might be reasonable, an exponential distribution doesn’t bind housing leverage at 100%. Thus, using an exponential distribution seems consistent with what one would expect--older households are far more likely to have no outstanding mortgages than being leveraged at 50% of the value of their home—while also not being bound between 0 and 100%.

Thus, while ordinary least squares regression could be utilized to try and predict housing leverage, a model that allows for an assumed distribution other than normal seems more reasonable.

Gauss-Markov Model

This model assumes that the error terms are distributed based on some function that is not necessarily normal, although still assuming that the variance is constant and the error terms are uncorrelated. While we know that these assumptions are not exactly true, they don’t appear to be too far off. However, given the discrete nature of some of the predictor variables, perhaps the ANCOVA model is better.

ANCOVA

The ANCOVA model allows for discrete and continuous variables to be included as regressor variables in a model. However, can an ANCOVA model assume the error terms are non-normally distributed (similar to the Gauss-Markov model)? The examples in class only showed models that assumed normal distributions.

Tobit

We also discussed briefly Tobit as a regression model that might be applicable when you are faced with an upper and lower bound constraint, possibly applicable here given the lower bound of 0% leverage and essentially 100% upper bound (even though very few individuals may have more than 100% leverage).

44115 about ANCOVA and unconditional means model.

Anonymous — Sat, 25 Apr 2009 21:45:59 GMT

In Marketing, some researchers use ‘cultural openness’ and degree of economic development of certain country to measure ‘willingness to buy global brand products’. Brands such as Coca-Cola, MTV, and Levis are the global brands which they have a standardized approach to their customers all over the world. Cultural openness can be defined as a person’s interest in and experience with foreign people, values, and cultures. I came up with one possible model (ANCOVA) for those constructs. Yij = mi + gx ij + eij where eij ~ iid N(o,s²) Yij=willingness to buy global brand productsmi=mean in each group (m0=mean in developing country and m1=mean in developed country)x= cultural openness i=0 (developed country) and 1(developing country) j= 1………n (number of individual in i –either developed or developing country category) Second model that I consider to use is an unconditional means model (mentioned in Judith D. Singer’s paper). Yij=m+aj+gij where aj~iid N (0, too) and gij~iid N (0,s²)Yij= willingness to buy global brand products m=linear combination of a grand mean a= GDP in jth country gij=random error of cultural openness associated with the ith individual in the jth country i=1, 2, 3, 4 ………k (number of individual in country q)j= 1, 2, 3, 4 ………q (number of country) Differences between two models are, 1) Unconditional means model used GDP (normally distributed) directly to indicate the degree of economic development of certain country whereas ANCOVA divided all countries into two groups and had a mean value for each group. 2) Second model (unconditional means model) assumes cultural openness is normally distributed while first model does not. I know both models are wrong (we never know the reality!). Nevertheless, do you think two models are possible options to predict marketing phenomenon mentioned above? And if they are, do you think they are interchangeable?

41941: Cross-sectional and Time-series Correlation in the Frozen Orange Juice Market

Anonymous — Sat, 25 Apr 2009 21:27:53 GMT

The field of finance frequently examines large panel data sets, and thus researchers are faced with the potential for the residuals to be correlated across firm and/or across time. On April 9^th, we discussed panel data sets and the potential for these types of residual correlations. Dr. Westfall referred to this correlation as “time-series correlation” (autocorrelation) and “cross-sectional correlation.” The purpose of this question is to better understand these two distinctly different types of correlation. Last semester in our seminar course, we read a paper that examined the frozen concentrated orange juice futures market to determine whether or not the futures contract price reflected fundamental values. For those non-finance majors, a futures contract is a contract between two parties for a particular commodity (in this case frozen concentrated orange juice) to be delivered sometime in the future at an agreed upon contract price. The interesting part about the frozen concentrated orange juice futures market is that the price of these contracts can be significantly influenced by the temperature in the Florida area, where most of the oranges are produced. More specifically, the supply of oranges is significantly altered when the temperature goes below freezing levels. The authors argue that the temperature, or Floridian weather, is a key factor in determining the returns on the frozen orange juice futures contract and the authors use two very different models. The first model the author’s explore is a basic linear model, which has been previously used by researchers. R_it = alpha_i + beta_1iW_t + E_it , where R_it is the close-to-close return, or [(P_t/ P_t-1) – 1], at time t for contract i, and W_t is the “contemporaneous realized minimum temperature” for time t. The second model is a second-order polynomial model, which the authors propose is a better model. R_it = alpha_i + beta_1i(max[0, W*-W_t]) + beta_2i(max[0,W*-W_t])² + E_it, where W* represents the critical temperature, which the authors set to 32 degrees Fahrenheit. Interestingly, the authors determine that the second model is better because “the linear model imposes the same relation between returns and temperature at temperatures both above and below freezing.” I would like to point out that the authors never examine/report the AIC, Press statistic or the adjusted R-squared, which we have extensively discussed in model selection portion of this course. Instead, the authors only compare the R-squared statistics. Is it okay to only compare the R-squared? From Dr. Westfall’s teaching I would conclude that they incorrectly determined the reasonableness of the model. However, the large sample size (1000+ observations) and the addition of only one extra parameter may imply that the adjusted R-squared is minimally different from the R-squared. Moreover, the authors never validated their assumption that the residuals where uncorrelated. Thus, I purpose the investigation of the potential of “time-series correlation” or “cross-sectional correlation” among the residual and I believe that the authors may have overlooked the potential for these correlations. Assuming all else constant, I think that that there should exist cross-sectional correlation between the returns because the contracts are all based on the same commodity – concentrated orange juice. Additionally, I believe that the time-series correlation may be less prevalent because returns from the prior day may or may not be correlated. For the sake of brevity, I would like to purpose the following pseudo panel data set.

Time (t) Contract Return (R) Temperature (W) Residual (E)

1 A 0.01 55 E_A1

1 B 0.015 55 E_B11 C 0.008 55 E_c12 A 0.03 57 E_A22 B 0.02 57 E_B2

2 C 0.02 57 E_C2

3 A - 0.13 29 E_A3

3 B - 0.15 29 E_B3

3 C - 0.10 29 E_C3

My question is based on the above panel data set, where time t is in days, contract type is either A, B, or C and temperature is in Fahrenheit. In class, Dr. Westfall said that the time series correlation was the residual correlation between, for example, E_A1and E_A2. Additionally, he said that the cross-sectional correlation was between, for example, E_A3 and E_B3. Let’s assume we run the regression of model 1 or model 2 by time. Once we get the residuals for this model, can we test the residuals for both types of correlation (cross-sectional, or time-series)? Or can we only test the residuals for cross-sectional correlation because the regression was done by time? If this is the case, then would I have to run the regression of the returns on the temperature, sorted by contract, in order to test the time-series correlation between the residuals? Additionally, if I find that the residuals do not exhibit either type of correlation, then is it okay to assume that the OLS estimates and standard errors are “reasonable”? Beyond looking at the graph (e.g. residuals against lag residuals) and “proc corr” is there any other way to test for the existence of correlation?

Citation:

Boudoukh, Jacob, Richardson, Matthew, Shen, YuQing, Whitelaw, Robert. 2007. Do asset prices reflect fundamentals? Freshly squeezed evidence from the OJ market. Journal of Financial Economics 83, 397 – 412.

49329 _ Random effect and Fixed effect

Anonymous — Sat, 25 Apr 2009 21:13:57 GMT

On April 14, you were talking about “mixed effect models” – models with both fixed and random effects, and then you wrote the one-way ANOVA models to demonstrate these effects. The models with fixed effect is postulated as follow.

Y_ij = mu_bar +(mu_i –mu_bar) +epsilon _ij = mu +apha_i + epsilon _ij,

where both mu and apha_i are fixed effect.

The model with random effect is expressed similarly Y_ij = mu +alpha_i + epsilon _ij, but now alpha_i is iid normally distrusted and independent of epsilon so it is the random effect of the class variable i the dependent variably Y. For example, i is the class variable demonstrating graduate major of student j at TTU.

On March 31, at the very beginning of the class, we discussed HW5, question 2, which is a small case study about the effect of GMAT on GPA. The null hypothesis is “Effect of GMAT is consistent across ethnic groups” and you wrote down the model as follow.

Y_ijkl = mu + alpha_i + beta_j + gama_k + lambda GMAT_ijkl + delta_i GMAT_ijkl + epsilon_ijkl

where i= 1 - #ethnic; j=1- # major; k = 1- # degree; and l = 1 - # students within each combination of ethnic, major and degree.

Even though we have not talked about random and fixed effects at the time we discuss this model, I think this model is also the mixed effect model with the mu and lambda GMAT_ijkl are fixed effects, and alpha_i , beta_j , gama_k and delta_i GMAT_ijkl are random effects. Am I right in understand this model this way? When we say random effect, it is the effect of class variable (e.g. ethnic group, major, degree) on GPA. But when we say fixed effect, then it is the fixed effect of what on GPA?

The mixed effect models that we are talking about here are in the ANOVA/ANCOVA forms. Does it have to be the case? Can we write in terms of regression models? For example, in finance, we are interested in the effect of some corporate events (e.g. merger and acquisition, seasonal equity issuing) on the stock returns – event study. One of the methods that we use is to estimate the abnormal returns (AR) on the stock after the announcement of the events. For example, if we want to see if the firm’s announcement of issuing equity will have negative effect on the stock returns, we can estimate the cumulative abnormal returns (CAR) for a short period of time around the announcement, say, from day -1 to day 1, where day 0 is the announcement day, for each firm that announce stock issuing. If the average CAR is negative at the significant level, then we reject the null hypothesis that stock issue announcement has no effect on the stock returns. In estimating the AR, we often use the Fama-French 3-factor regression model postulated as follow:

R_ij = R_rfj+ alpha_ij+ beta1* (R_mj – R_rfj ) + beta2*SMB_ij +beta3*HML_ij + epsilon_ij , (1)

In order to run regression, the model is usually expressed as

R_ij - R_rfj = alpha_ij+ beta1* (R_mj – R_rfj ) + beta2*SMB_ij +beta3*HML_ij + epsilon_ij , (2)

Where R_i is the return on stock i in time j (j= -1, 0, 1)

R_rfj is market risk free rate (usually 3 month T-bill rate)

R_mj is return on the market index

SMB_ij is small minus big, capturing the size effect of the firms

HML_ij is high minus low, capturing the growth effect of the firms.

Thus, the intercept or alpha is interpreted as the abnormal return of the firms due to the event.

My question is, regardless of the form (1) or (2), can I interpret R_rfj as the fixed effect and alpha_ij as the random effect of the event on the return of the stock?

44798 Clustered data, BLUP and endogeneity problem

Anonymous — Sat, 25 Apr 2009 20:51:00 GMT

When data are clustered with error terms correlated within a cluster and uncorrelated between clusters, Dr. Westfall showed that the standard error of OLS overestimated the standard deviation of true model, leading to fail to reject the null of X = 0 while ML based on GLS that allow correlation within a cluster correctly reject the null of X=0. Furthermore, we learned that with the clustered data the sample size can vary. In this situation, it is important to account for the sample size using BLUP (random effect). Otherwise, a cluster with the small number of sample size with little variation might elicit the best result even if that is not true. For example, the department’s teaching ranking by simple mean was not credible because the best one from agricultural communication had only one sample size. However, when the simulation accounted for the sample size by BLUP, agricultural communication was ranked as 38^th which is much more reliable result.

In the research of corporate governance, researchers examine the effect of good corporate governance on firm performance. In simplest version, proxy for good corporate governance is the size of board and firm performance is measured by ROA.

That is, ROA = b0+ b1boardsize + industry + e. The board size is likely to vary depending on the industries and firms within the same industry are likely to be clustered in this model. In other words, the error terms are correlated within an industry but uncorrelated between industries. For instance, the number of boards in manufacturing industry, on average, is 10 while the number of boards in finance industry, on average, 5. Besides, things other than board size that affect the ROA of firms within a finance industry can be correlated each other. In this situation I can apply the following code. (I don’ have real data. This is virtual one)

proc mixed data= governance covtest; class industry; model ROA = board size / s; random industry /s;

run; quit;

However, another problem that concerns me is endogeneity bias which is most common problem in corporate governance. I assume that firm performance can be different depending on the board size. The opposite, however, can be true. As the firm’s profit increases or decreases, board size can change. In equation form, ROA = b0+ b1boardsize + industry + e. Board size = b0 + b1ROA + industry + e. These equations leads me to consider simultaneous equation model which we covered last class. Using OLS will bring biased estimates. Thus, I should use instrumental variable in order to obtain unbiased estimates. I can apply the following code, then.

proc syslin data= governance 2sls;

   instruments industry;

   eq1: model ROA = BOARDSIZE INDUSTRY;

   eq2: model BOARDSIZE = ROA INDUSTRY;

run;

Do you think that industry here is right instrument variable? I am very confused with instrument variable. What IV might be possibly good candidate in this situation?

According to the book, IV must be correlated with board size but can not be correlated with error term. I don’t understand this last sentence clearly. Does that mean that IV has to be correlated with board size but not correlated with ROA and industry?

47761, Application of School effects model and individual growth models on corporate governance

Anonymous — Sat, 25 Apr 2009 20:35:56 GMT

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This summer I am doing a research on S&P 500 firms’ board members. One of the hypotheses that I am trying to prove is that after Sarbanes-Oxley Act (SOX), the demand for board members with financial and legal expertise increased. Therefore I claim that demand for CFOs sitting on other firms’ boards increased (implicitly assuming that CFO is expert in financial matters). I have not come up with a proxy position for legal expertise yet although I have data on executives’ educational background. So, for example, if a CFO of Microsoft were sitting at two firm’s boards, say Intel and Dell, now he or she might be sitting on more firms’ board.

Inspired by Judith D. Singer’s paper, I would like to model this by using either School effects model , if possible. My dependent variable would be CFOs, and my independent variable would be the number of outside board seats the executive holds. In Singer’s paper, page 326, the model is

Y_i,j = µ + α _i + r_i,j (1)

where Y_i,j is the score of jth pupil at the ith school, µ is the average of all scores (fixed effect), α _i is the variation between school means ( random effect), and r_i,j is the variation among students within schools.

My first question, can I use this model, by replacing the pupils with, say, CFO, and test scores with number of outside board seats they hold? My model would look like:

Y_i,j = µ + α _i + r_i,j (2)

where Y_i,j is the number of outside board seats jth CFO holds at the ith company, µ is the average of all number of outside board seats (fixed effect), α _i is the variation between company means ( random effect), and r_i,j is the variation among CFOs within companies. The random effect part might come from different companies have different policies for CFOs holding outside directorships, or the additional tasks that a particular CFO holds in his own board. Unlike the Singer’s model, my dependent variable is categorical data but the Proc Mixed does not restrict me. Can I use the example that we covered in class where we compared rankings of teaching in various TTU majors using OLS vs. BLUPs as a starting point?

My second question is related with another hypothesis that I would like to test. I claim that because of the recent financial crisis, executives are abandoning their outside board

 memberships and trying to focus on their own companies. Do you think Singer’s individual growth model is appropriate to model this? Thank you in advance.

45967 Variable selection and variance-bias trade off.

Anonymous — Sat, 25 Apr 2009 18:44:29 GMT

45967 Variable selection and variance-bias trade off.My question is whether I have found a good model for my research.The purpose of my research is to investigate the relationship of the wife and husband in household risk management. I used similar X variables as previous literature. The independent variables are as follows: dual income, owned house, number of household member, stage of family life cycle depends on first baby, education of husband, knowledge of financial management, interest of financial management, health condition of husband and wife, total income, net asset, expenditure, and debt. And dependent variable is the wife and husband role division in household risk management. These data have some categorical variables and numeric variables. Dependent variable: The dependent variable is risk management. A higher number indicates that the husband is more engaged in risk management than the wife, (all by wife), 2(mostly by wife), 3(both similarly), 4(mostly by husband), 5(all by husband). Categorical variables:Dual is a 0/1 variable (single income / dual income)House is a 0/1 variable (not owned house / owned house) Eldest is 1 (eldest not attending school), 2 (eldest is elementary grade), 3 (eldest is junior high school), 4 (eldest is college student), 5 (eldest graduated from college) Knowledge is 1 (wife have more financial knowledge), 2(husband have more financial knowledge), 3(wife and husband have a same financial knowledge) Manage is 1 (wife have more financial interest), 2(husband have more financial interest), 3(wife and husband have a same financial interest) Education is 1 (junior high school), 2(community college), 3(college) Numeric variables:Size is number of household memberHhealth (health condition of husband) is self test and greater number is stronger Health (health condition of wife ) is self test and greater number is stronger Income is total household income (husband income + wife income)Net asset is total asset – total debtExpenditure is monthly expenditure
I assumed a linear model. Full regression model is Risk_i = β₀ + β₁Dual1_i + β₂Size_i + β₃House_1i + β₄Eldest_2i + β₅Eldest_3i+ β₆Eldest_4i + β₇Eldest_5i + β₈Edu_2i + β₉Edu_3i + β₁₀ Hhealth_i + β₁₁Health_i+ β₁₂Manage_1i + β₁₃Manage_2i + β₁₄Knowledge_1i + β₁₅Knowledge_2i + β₁₆Income_i + β₁₇Netasset_i + β₁₈Expenditure_i + ε_i

To find out if I had a multicollinearity problem I tested correlation between income, net asset and expenditure. I took off the net asset variable and expenditure variable from the full model because of they were significantly correlated with each other.

Figure 1. Multiply regression without net asset and expenditure

The REG Procedure Model: MODEL1 Dependent Variable: risk Number of Observations Read 459 Number of Observations Used 459 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 16 4730.76988 295.67312 9.44 <.0001 Error 442 13838 31.30859 Corrected Total 458 18569 Root MSE 5.59541 R-Square 0.2548 Dependent Mean 17.80392 Adj R-Sq 0.2278 Coeff Var 31.42795 Parameter Estimates Parameter Standard Standardized Variable DF Estimate Error t Value Pr > |t| Estimate Intercept 1 15.90172 2.18227 7.29 <.0001 0 dual 1 -0.38324 0.56263 -0.68 0.4961 -0.03002 size 1 0.31982 0.32631 0.98 0.3276 0.04980 house1 1 0.11698 0.62081 0.19 0.8506 0.00888 hedu2 1 -0.44513 0.85096 -0.52 0.6012 -0.02559 hedu3 1 0.07779 0.66243 0.12 0.9066 0.00606 eldest2 1 -1.35288 0.84503 -1.60 0.1101 -0.08300 eldest3 1 -1.72871 0.93581 -1.85 0.0654 -0.09536 eldest4 1 -2.60658 0.93485 -2.79 0.0055 -0.14904 eldest5 1 -1.30686 1.01514 -1.29 0.1986 -0.06827 hhealth 1 1.19654 0.44424 2.69 0.0073 0.13097 health 1 -0.51816 0.41784 -1.24 0.2156 -0.05995 manage1 1 -2.39296 0.74306 -3.22 0.0014 -0.18712 manage2 1 2.20850 0.80417 2.75 0.0063 0.14776 knowl1 1 -1.55472 0.79788 -1.95 0.0520 -0.11089 knowl2 1 1.87069 0.71812 2.61 0.0095 0.14359 trincome 1 -0.00066950 0.00067574 -0.99 0.3223 -0.04331 According to multiple regression analysis, only eldest4, manage1, manage2 and knowl2 variables have statistically significant effects on the risk management of household. To find which variables to include in my final model I used the stepwise analysis. According to stepwise analysis six variables could be predict division of household risk management between the wife and husband in figure 2. Figure 2. Stepwise resultscorrelation matrix of X variables 50 The REG Procedure Model: MODEL1 Dependent Variable: risk Summary of Stepwise Selection Variable Variable Number Partial ModelStep Entered Removed Vars In R-Square R-Square C(p) F Value Pr > F 1 manage1 1 0.1606 0.1606 42.8257 87.46 <.0001 2 knowl2 2 0.0458 0.2064 17.6604 26.32 <.0001 3 hhealth 3 0.0107 0.2172 13.2946 6.24 0.0129 4 manage2 4 0.0099 0.2270 9.4472 5.79 0.0165 5 eldest4 5 0.0072 0.2342 7.1778 4.26 0.0396 6 knowl1 6 0.0075 0.2417 4.7314 4.47 0.0351I put six variables into my model to predict the division of household risk management between the wife and husband. I used reduced regression model:
Risk_i = β₀ + β₁Eldest_4i + β₂ Hhealth_i + β₃Manage_1i + β₄Manage_2i + β₅Knowledge_1i + β₆Knowledge_2i + ε_i

The results show me that the person with greater interest and knowledge in financial management tended to have more responsiblity in risk management.

Figure 3. Reduced model

The REG Procedure Model: MODEL1 Dependent Variable: risk Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 6 4488.71075 748.11846 24.02 <.0001 Error 452 14080 31.15145 Corrected Total 458 18569 Root MSE 5.58135 R-Square 0.2417 Dependent Mean 17.80392 Adj R-Sq 0.2317 Coeff Var 31.34898 Parameter Estimates Parameter Standard Standardized Variable DF Estimate Error t Value Pr > |t| Estimate Intercept 1 14.67081 1.51397 9.69 <.0001 0 eldest4 1 -1.61778 0.72069 -2.24 0.0253 -0.09250 hhealth 1 0.99708 0.37658 2.65 0.0084 0.10913 manage1 1 -2.36294 0.72468 -3.26 0.0012 -0.18478 manage2 1 2.13402 0.79423 2.69 0.0075 0.14277 knowl1 1 -1.65441 0.78261 -2.11 0.0351 -0.11800 knowl2 1 1.83752 0.70624 2.60 0.0096 0.14105

Question: In the class, we learned about the variance – bias trade off. In my research, I chose the reduced model to predict the division of household risk management between the wife and husband instead of the full model which was introduced first. Can I say that I am willing to accept the bias of less complex model (reduced model) to offset the variance and was stepwise analysis a good way to reduce the model?

48828 - Insignificant Interactions in Experimental Research

Anonymous — Sat, 25 Apr 2009 19:34:38 GMT

Intro:
I am submitting a experimental research proposal in about a week. I don't have any data to analyze; I only have a theory of what I believe will happen in an experiment which I have not yet conducted. In the experiment, participants will be given one of two reports (treatment or control) and then asked to revise a previously expressed belief. Revision of a previously held belief, or the extent to which one is willing to revise their belief can be seen as "interpendence" or "lack of independence" in this case because the participants will be adjusting from a personal belief after given information about a group's belief (in the treatment report).

To simplify, participants in this study will be exposed to either "Treatment A" or a will be in the control group. I believe that participants exposed to "Treatment A" will exhibit more independence than those in the control group.

Controlling for Predisposition to Act Independently:
In the experimental design I am relying on random assignment and trusting that most things affecting a person's independence will be distributed about evenly across the two groups. However, people vary in their predisposition to act independently, and this characteristic can be captured in a survey. I want to make sure that I control for this individual trait in my model. In other words, I am not trusting random design to distribute this trait evenly among the two groups. It is too important and the trait is easily captured with an established survey.

Interaction between the Control and the Treatment:
Similar studies would suggest that people who score higher on the survey will be more sensitive to Treatment A. Thus, I have constructed the following regression model;

Y = B0 + B1(Td) + B2(SCORE) + B3(Td*SCORE) + error.

Where: Y = distribution of independence expressed in the estimate revision
Td = indicator variable for treatment A
SCORE = The score on the survey (remember - this measures a person's predisposition to act independently)

I am mainly interested in whether the treatment has an effect on Y. If I am able to say this in my future paper then I think people who read it would learn something. Also, I believe that the model nicely communicates the process which I believe takes place in american corporations.

The Expected Limitation of the Study:
I want to generalize my study to "corporate employees". However, my participants will be college students at Texas Tech University. I do not expect that the class will be as diverse as the corporate workforce. But, I believe that I have a good argument in place that justifies the use of students in this experiment. Specifically, I think that - in the proposal - I have sufficiently argued that I can generalize the effects of the treatment to corporate employees. However, I am not confident that I will be able to find an interaction effect between the treatment and the survey score in my experimental setting. The reason is that I believe that the predisposition of people to act independently varies more in the corporate workforce than it will in TTU students. In other words, I don't believe that I will have enough variance in the survey scores of students to detect a significant interaction in my experimental setting. However, I do believe that this interaction is real, i.e. it is part of the process which I am interested in (the behavior patterns of corporate employees).

My question:

The technical part of my question is: Should I include the interaction term when submitting the proposal?

The more philosophical part of my question is: When developing a theory in conjunction with an experimental design, should the researcher take the expected limitations of the design into consideration when developing the model? So, would I be better off developing a model for the process to which I wish to generalize and then explaining that expected interactions are not significant because of an experimental limitation. -or- Should I develop the more simple model based on what I believe will happen in the experimental setting and then explain how the simple model corresponds to the real-life situation of interest? Which makes more sense scientifically?

Or, restated but possibly more straight forward: Should I try to model the actual process (behavior patterns of corporate employees) or the process which I expect to see in the lab (behavior patterns of students)?

46724 Model fit and academic research.

Anonymous — Sat, 25 Apr 2009 07:33:12 GMT

Faig and Shum (2002) propose the following model to predict safe and liquid assets in a portfolio:

7 8Cash_i= β₀+ β₁Age_i + β₂Age²_i + ∑ β_3jX_ji+ ∑ β_4kD_ki + ε_i. j=1 k=1Cash is the percentage of safe and liquid assets in the portfolio. The seven control variables (the X_j’s) besides age and age squared are listed in the table below. The eight dummy variables for saving motives (D_k’s) are also listed in the table along with the significance results from the regression.

		Statistically
		Significant
	Intercept	Yes
	Age	Yes
	Age²	Yes
X₁	Financial Net Worth	No
X₂	Financial Net Worth²	No
X₃	Relative Housing Value	Yes
X₄	Relative Inv. Real Estate	Yes
X₅	Risk Attitude	Yes
X₆	Relative Business Value	Yes
X₇	Log of Labor Income	Yes
D₁	Education	No
D₂	Invest in own Home	Yes
D₃	Household Purchases	No
D₄	Travel	No
D₅	Invest in own Business	Yes
D₆	Retirement	Yes
D₇	Emergency	No
D₈	Living expenses and bills	No

The model presented has several insignificant control and dummy variables. In this study and in most studies I have read there is not any mention of model fit. They do however discuss the theoretical reason for the inclusion of the variables. If the authors for this paper tested several models and chose the model with the best fit, with their available data, based on AIC they likely would have had fewer variables included in their model. They do present more than one model in this paper but there is no discussion of model fit as the reason for additional models. It seems that the discussion of the variables is more important than attempting to find a model that is as close to the true model as possible. When selecting variables I understand why they may not want to drop variables that have a strong theoretical reason for being included because when future studies are done with different data sets those variables may actually be significant and make for a model that fits better to that particular data. However it seems that they should probably include two models, one model that is suggested prior to pairing down any variables and one model that strives to fit the data as best as possible. Taking this latter approach would allow for discussion of variables that are not included in the model that was chosen using AIC and by suggesting the model with the best fit you likely would have a model that is closer to the true model. Does this seem to be a reasonable approach? If so why is it not done more often? Faig, Miquel, and Pauline Shum, 2002, Portfolio Choice in the Presence of Personal Illiquid Projects, Journal of Finance 57, 303-328.

48998 Regression versus ANOVA

Anonymous — Sun, 26 Apr 2009 03:27:17 GMT

Dr. Westfall,

q = m_cs – m_pc

The statistical model is represented by the following:

Y_ijk ~ N (m_ij, s²) independent

But I want to control for “attitudes toward management classes,” which is also measured on a 5-point Likert scale, because this factor could be a confounding variable.

So this ANOVA model be written as the following: Y_ijk = m_ij + z_ij+e_ijk, where Z_ij is the “current attitudes toward a management class.”

For the same study, the regression model would be:

48545 - Snowball Sampling & Correlation of Error Term Concerns

Anonymous — Sun, 26 Apr 2009 02:16:41 GMT

Please see my other post with the same name for my question. This post is just to include my ID (48545) which I accidentally left out of the subject line on the last post.

Thanks,