1 Lecture Ten

2 Lecture Part I: Regression –properties of OLS estimators –assumptions of OLS –pathologies of OLS –diagnostics for OLS Part II: Experimental Method

3 Properties of OLS Estimators Unbiased: Note: y(i) = a + b*x(i) + e(i) And summing over observations i and dividing by n: Recall, the estimator for the slope is:

4 And substituting in this expression for the estimator, the expression for And taking expectations Note:

5 So The dispersion in the estimate for the slope depends upon unexplained variance, and inversely on the dispersion in x. the estimate, the unexplained mean square, is used for the variance of e.

6 Other Properties of Estimators Efficiency: makes optimum use of the sample information to obtain estimators with minimum dispersion Consistency: As the sample size increases the estimator approaches the population parameter

7 Outline: Regression The Assumptions of Least Squares The Pathologies of Least Squares Diagnostics for Least Squares

Assumptions Expected value of the error is zero, E[e]= 0 The error is independent of the explanatory variable, E{e [x-Ex]}=0 The errors are independent of one another, E[e(i)e(j)] = 0, i not equal to j. The variance is homoskedatic, E[e(i)] 2 =E[e(j)] 2 The error is normal with mean zero and variance sigma squared,

Error Variable: Required Conditions The error  is a critical part of the regression model. Four requirements involving the distribution of  must be satisfied. –The probability distribution of  is normal. –The mean of  is zero: E(  ) = 0. –The standard deviation of  is   for all values of x. –The set of errors associated with different values of y are all independent.

The Normality of  From the first three assumptions we have: y is normally distributed with mean E(y) =  0 +  1 x, and a constant standard deviation   From the first three assumptions we have: y is normally distributed with mean E(y) =  0 +  1 x, and a constant standard deviation     0 +  1 x 1  0 +  1 x 2  0 +  1 x 3 E(y|x 2 ) E(y|x 3 ) x1x1 x2x2 x3x3  E(y|x 1 )  The standard deviation remains constant, but the mean value changes with x

11 Pathologies Cross section data: error variance is heteroskedatic. Example, could vary with firm size. Consequence, all the information available is not used efficiently, and better estimates of the standard error of regression parameters is possible. Time series data: errors are serially correlated, i.e auto-correlated. Consequence, inefficiency.

12 Pathologies ( Cont. ) Explanatory variable is not independent of the error. Consequence, inconsistency, i.e. larger sample sizes do not lead to lower standard errors for the parameters, and the parameter estimates (slope etc.) are biased. The error is not distributed normally. Example, there may be fat tails. Consequence, use of the normal may underestimate true 95 % confidence intervals.

13 Pathologies (Cont.) Multicollinearity: The independent variables may be highly correlated. As a consequence, they do not truly represent separate causal factors, but instead a common causal factor.

Regression Diagnostics - I The three conditions required for the validity of the regression analysis are: –the error variable is normally distributed. –the error variance is constant for all values of x. –The errors are independent of each other. How can we diagnose violations of these conditions?

15 Residual Analysis Examining the residuals (or standardized residuals), help detect violations of the required conditions. Example 18.2 – continued: –Nonnormality. Use Excel to obtain the standardized residual histogram. Examine the histogram and look for a bell shaped. diagram with a mean close to zero.

16 Diagnostics ( Cont. ) Multicollinearity may be suspected if the t- statistics for the coefficients of the explanatory variables are not significant but the coefficient of determination is high. The correlation between the explanatory variable can then be calculated. To see if it is high.

17 Diagnostics Is the error normal? Using EViews, with the view menu in the regression window, a histogram of the distribution of the estimated error is available, along with the coefficients of skewness and kurtosis, and the Jarque-Bera statistic testing for normality.

18 Diagnostics (Cont.) To detect heteroskedasticity: if there are sufficient observations, plot the estimated errors against the fitted dependent variable

Heteroscedasticity When the requirement of a constant variance is violated we have a condition of heteroscedasticity. Diagnose heteroscedasticity by plotting the residual against the predicted y The spread increases with y ^ y ^ Residual ^ y

20 Homoscedasticity When the requirement of a constant variance is not violated we have a condition of homoscedasticity. Example continued

21 Diagnostics ( Cont.) Autocorrelation: The Durbin-Watson statistic is a scalar index of autocorrelation, with values near 2 indicating no autocorrelation and values near zero indicating autocorrelation. Examine the plot of the residuals in the view menu of the regression window in EViews.

22 Non Independence of Error Variables –A time series is constituted if data were collected over time. –Examining the residuals over time, no pattern should be observed if the errors are independent. –When a pattern is detected, the errors are said to be autocorrelated. –Autocorrelation can be detected by graphing the residuals against time.

23 Patterns in the appearance of the residuals over time indicates that autocorrelation exists Time Residual Time Note the runs of positive residuals, replaced by runs of negative residuals Note the oscillating behavior of the residuals around zero. 00 Non Independence of Error Variables

24 Fix-Ups Error is not distributed normally. For example, regression of personal income on explanatory variables. Sometimes a transformation, such as regressing the natural logarithm of income on the explanatory variables may make the error closer to normal.

25 Fix-ups (Cont.) If the explanatory variable is not independent of the error, look for a substitute that is highly correlated with the dependent variable but is independent of the error. Such a variable is called an instrument.

26 Data Errors: May lead to outliers Typos may lead to outliers and looking for ouliers is a good way to check for serious typos

27 Outliers An outlier is an observation that is unusually small or large. Several possibilities need to be investigated when an outlier is observed: –There was an error in recording the value. –The point does not belong in the sample. –The observation is valid. Identify outliers from the scatter diagram. It is customary to suspect an observation is an outlier if its |standard residual| > 2

The outlier causes a shift in the regression line … but, some outliers may be very influential An outlier An influential observation

29 Procedure for Regression Diagnostics Develop a model that has a theoretical basis. Gather data for the two variables in the model. Draw the scatter diagram to determine whether a linear model appears to be appropriate. Determine the regression equation. Check the required conditions for the errors. Check the existence of outliers and influential observations Assess the model fit. If the model fits the data, use the regression equation.

30 Part II: Experimental Method

31 Outline Critique of Regression

32 Critique of Regression Samples of opportunity rather than random sample Uncontrolled Causal Variables –omitted variables –unmeasured variables Insufficient theory to properly specify regression equation

33 Experimental Method: # Examples Deterrence Aspirin Miles per Gallon

34 Deterrence and the Death Penalty

35 Isaac Ehrlich Study of the Death Penalty: _Homicide Rate Per Capita _Control Variables _probability of arrest _probability of conviction given charged _Probability of execution given conviction _Causal Variables _labor force participation rate _unemployment rate _percent population aged years _permanent income _trend

Long Swings in the Homicide Rate in the US: Source: Report to the Nation on Crime and Justice

Ehrlich Results: Elasticities of Homicide with respect to Controls Source: Isaac Ehrlich, “The Deterrent Effect of Capital Punishment

38 Critique of Ehrlich by Death Penalty Opponents _Time period used: _period of declining probability of execution _Ehrlich did not include probability of imprisonment given conviction as a control variable _Causal variables included are unconvincing as causes of homicide

39 United States Bureau of Justice Statistics

40 Experimental Method Police intervention in family violence

41 United States Bureau of Justice Statistics

42 United States Bureau of Justice Statistics

43 Police Intervention with Experimental Controls _A 911 call from a family member _the case is randomly assigned for “treatment” _A police patrol responds and visits the household _police calm down the family members _based on the treatment randomly assigned, the police carry out the sanctions

44 Why is Treatment Assigned Randomly? _To control for unknown causal factors _assign known numbers of cases, for example equal numbers, to each treatment _with this procedure, there should be an even distribution of difficult cases in each treatment group

call (characteristics of household Participants unknown) Random Assignment code blue code gold patrol responds settles the household verbally warn the husbandtake the husband to jail for the night

46 Experimental Method: Clinical Trials Doctors Volunteer Randomly assigned to two groups treatment group takes an aspirin a day the control group takes a placebo (sugar pill) per day After 5 years, 11,037 experimentals have 139 heart attacks (fatal and non fatal) p E = after 5 years, controls have 239 heart attacks, p c =

Conclusions from the Clinical Trials Hypotheses: H 0 : p C = p E, or p C - p E = 0.; H a : (p C - p E ) 0. Statistic:Z = [ C - E ) – (p C - p E )]/  ( p C - p E ) recall, from the variance for a proportionSE SE( C - E )={[ c (1- c )]/n c + [ E (1- E )]/n E } 1/2 { [0.={[0217 ( )/ 11,034] + [ ( 1 – )/ 11,039} 1/2 = , so z = ( )/ z= 5.2

48 Experimental Method Experimental Design: Paired Comparisons comparing mileage for two different brands of gasoline control for variation in car and driver by having each cab use both gasolines. Each cab is called a block in the experimental design control for weather, traffic, and other factors by assigning different days and times to each cab.

Table 1: Miles Per Gallon for Brand A and Brand B

50 Test Whether the Difference Between Gasolines is Zero: H 0 : diff = 0, H a : diff not zero t-stat = (sample difference - zero)/(smpl. std. dev/n) t-stat = -0.60/(0.61/10 1/2 ) = /0.190 =

51 Midterm 2000.(15 points) The following table shows the results of regressing the natural logarithm of California General Fund expenditures, in billions of nominal dollars, against year beginning in 1968 and ending in A plot of actual, estimated and residual values follows. –.How much of the variance in the dependent variable is explained by trend? –.What is the meaning of the F statistic in the table? Is it significant? –.Interpret the estimated slope. –.If General Fund expenditures was $ billion in California for fiscal year , provide a point estimate for state expenditures for

52 Cont. –A state senator believes that state expenditures in nominal dollars have grown over time at 7% a year. Is the senator in the ballpark, or is his impression significantly below the estimated rate, using a 5% level of significance? –If you were an aide to the Senator, how might you criticize this regression?

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