Exam 10: Regression With Panel Data

In the panel regression analysis of beer taxes on traffic deaths, the estimation period is 1982-1988 for the 48 contiguous U.S. states. To test for the significance of time fixed effects, you should calculate the F -statistic and compare it to the critical value from your $F _ { q , \infty }$ distribution, which equals (at the 5 % level)

You want to find the determinants of suicide rates in the United States.To investigate the issue, you collect state level data for ten years.Your first idea, suggested to you by one of your peers from Southern California, is that the annual amount of sunshine must be important.Stacking the data and using no fixed effects, you find no significant relationship between suicide rates and this variable.(This is good news for the people of Seattle.)However, sorting the suicide rate data from highest to lowest, you notice that those states with the lowest population density are dominating in the highest suicide rate category.You run another regression, without fixed effect, and find a highly significant relationship between the two variables.Even adding some economic variables, such as state per capita income or the state unemployment rate, does not lower the t-statistic for the population density by much.Adding fixed entity and time effects, however, results in an insignificant coefficient for population density. (a)What do you think is the cause for this change in significance? Which fixed effect is primarily responsible? Does this result imply that population density does not matter?

A researcher investigating the determinants of crime in the United Kingdom has data for 42 police regions over 22 years.She estimates by OLS the following regression $\ln ( \mathrm { cmrt } ) _ { i t } = \alpha _ { i } + \phi _ { t } + \beta _ { 1 } \text { unrtm } _ { i t } + \beta _ { 2 } \text { proyth } _ { i t } + \beta _ { 3 } \ln ( p p ) _ { i t } + u _ { i t } ; i = 1 , \ldots , 42 , t = 1 , \ldots , 22$ where $\mathrm { cmrt }$ is the crime rate per head of population, unrtm is the unemployment rate of males, proyth is the proportion of youths, $p p$ is the probability of punishment measured as (number of convictions)/(number of crimes reported). $\alpha$ and $\phi$ are area and year fixed effects, where $\alpha _ { i }$ equals one for area $i$ and is zero otherwise for all $i$ , and $\phi _ { t }$ is one in year $t$ and zero for all other years for $t = 2 , \ldots , 22 . \phi _ { 1 }$ is not included. (a) What is the purpose of excluding $\phi _ { 1 }$ ? What are the terms $\alpha$ and $\phi$ likely to pick up? Discuss the advantages of using panel data for this type of investigation.

In the Fixed Time Effects regression model, you should exclude one of the binary variables for the time periods when an intercept is present in the equation

The Fixed Effects regression model A)has n different intercepts. B)the slope coefficients are allowed to differ across entities, but the intercept is "fixed" (remains unchanged). C)has "fixed" (repaired)the effect of heteroskedasticity. D)in a log-log model may include logs of the binary variables, which control for the fixed effects.

In the panel regression analysis of beer taxes on traffic deaths, the estimation period is 1982-1988 for the 48 contiguous U.S. states. To test for the significance of time fixed effects, you should calculate the F -statistic and compare it to the critical value from your $F _ { q , \infty }$ distribution, where q equals

In the panel regression analysis of beer taxes on traffic deaths, the estimation period is 1982-1988 for the 48 contiguous U.S. states. To test for the significance of entity fixed effects, you should calculate the F -statistic and compare it to the critical value from your $F _ { q , \infty }$ distribution, where q equals

In the Fixed Effects regression model, using (n - 1)binary variables for the entities, the coefficient of the binary variable indicates a. the level of the fixed effect of the $i ^ { \text {th } }$ entity. b. will be either 0 or 1 . c. the difference in fixed effects between the $i ^ { \text {th } }$ and the first entity. d. the response in the dependent variable to a percentage change in the binary variable.

Consider estimating the effect of the beer tax on the fatality rate, using time and state fixed effect for the Northeast Region of the United States (Maine, Vermont, New Hampshire, Massachusetts, Connecticut and Rhode Island)for the period 1991-2001. If Beer Tax was the only explanatory variable, how many coefficients would you Need to estimate, excluding the constant?

Indicate for which of the following examples you cannot use Entity and Time Fixed Effects: a regression of

If you included both time and entity fixed effects in the regression model which includes a constant, then a. one of the explanatory variables needs to be excluded to avoid perfect multicollinearity. b. you can use the "before and after"' specification even for $T > 2$ . c. you must exclude one of the entity binary variables and one of the time binary variables for the OLS estimator to exist. d. the OLS estimator no longer exists.

Your textbook modifies the four assumptions for the multiple regression model by adding a new assumption. This represents an extension of the cross-sectional data case, where errors are uncorrelated across entities. The new assumption requires the errors to be uncorrelated across time, conditional on the regressors as well $\left( \operatorname { cov } \left( u _ { i t } , u _ { i s } \mid X _ { i t } , X _ { i s } \right) = 0 \right.$ for $t \neq s$ . (a)Discuss why there might be correlation over time in the errors when you use U.S.state panel data.Does this mean that you should not use OLS as an estimator?

Consider the time and entity fixed effect model with a single explanatory variable $Y _ { i t } = \beta _ { 0 } + \beta _ { 1 } X _ { i t } + \gamma _ { 2 } D 2 _ { i } + \ldots + \gamma _ { n } D n _ { i } + \delta _ { 2 } B 2 _ { t } + \ldots + \delta _ { T } B T _ { t } + u _ { i t } ,$ Assume that you had estimated the above equation by OLS. Typically the coefficients for the entity and time binary variables are not reported. Can you think of situations where the pattern of these coefficients might be of interest? What could you do, for example, if you had a strong theoretical justification for believing that a few macroeconomic variables had an effect on $Y _ { i t }$ ?

In Sports Economics, production functions are often estimated by relating the winning percentage of teams (Y)to inputs indicating performance in certain aspects of the game. However, this omits the quality of management.Assume that you could measure the quality of pitching and hitting by a single index L, and that managerial ability is represented by M, which is assumed to be constant over time.The production function would then be specified as follows: $Y _ { i t } = \beta _ { 0 } + \beta _ { 1 } L _ { i t } + \beta _ { 2 } M _ { i } + u _ { i t }$ where i is an index for the baseball team, and t indexes time and all variables are in logs. (a) Assume that managerial ability is unobservable but is positively related, in a linear way, to $L$ . Explain why the OLS estimator $\widehat { \beta } _ { 1 }$ is inconsistent in the case of a single crosssection, i.e., if you attempt to estimate the above regression for a single year. Do you expect this coefficient to over- or under-estimate $\beta _ { 1 }$ ?

Two authors published a study in 1992 of the effect of minimum wages on teenage employment using a U.S.state panel.The paper used annual observations for the years 1977-1989 and included all 50 states plus the District of Columbia.The estimated equation is of the following type $E _ { i t } = \beta _ { 0 } + \beta _ { 1 } \left( M _ { i t } / W _ { i t } \right) + \gamma _ { 2 } D 2 _ { i } + \ldots + \gamma _ { n } D 51 _ { i } + \delta _ { 2 } B 2 _ { t } + \ldots + \delta _ { \mathrm { T } } \mathrm { B } 13 _ { \mathrm { t } } + u _ { i t }$ where E is the employment to population ratio of teenagers, M is the nominal minimum wage, and W is average wage in the state.In addition, other explanatory variables, such as the prime-age male unemployment rate, and the teenage population share were included. (a)Briefly discuss the advantage of using panel data in this situation rather than pure cross sections or time series.

The difference between an unbalanced and a balanced panel is that

Consider the case of time fixed effects only, i.e., $Y _ { i t } = \beta _ { 0 } + \beta _ { 1 } X _ { i t } + \beta _ { 3 } S _ { t } + u _ { i t } ,$ First replace $\beta _ { 0 } + \beta _ { 3 } S _ { t }$ with $\phi _ { t }$ . Next show the relationship between the $\phi _ { t }$ and $\delta _ { i }$ in the following equation $Y _ { i t } = \beta _ { 0 } + \beta _ { 1 } X _ { i t } + \delta _ { 2 } B 2 _ { t } + \ldots + \delta _ { T } B T _ { t } + u _ { i t }$ where each of the binary variables B2, …, BT indicates a different time period.Explain in words why the two equations are the same.Finally show why there is perfect multicollinearity if you add another binary variable B1.What is the intuition behind the fact that the OLS estimator does not exist in this case? Would that also be the case if you dropped the intercept?

Consider a panel regression of unemployment rates for the G7 countries (United States, Canada, France, Germany, Italy, United Kingdom, Japan)on a set of explanatory Variables for the time period 1980-2000 (annual data).If you included entity and time Fixed effects, you would need to specify the following number of binary variables:

Consider the following panel data regression with a single explanatory variable $Y _ { i t } = \beta _ { 0 } + \beta _ { 1 } X _ { i t } + u _ { i t } .$ In each of the examples below, you will be adding entity and time fixed effects.Indicate the total number of coefficients that need to be estimated. (a)The effect of beer taxes on the fatality rate, annual data, 1982-1988, nine U.S.regions (New England, Pacific, Mid-Atlantic, East North Central, etc.).

The notation for panel data is $\left( X _ { i t } , Y _ { i t } \right) , i = 1 , \ldots , n \text { and } t = 1 , \ldots , T$ because