Appendix D INTRODUCTION TO BOOTSTRAP ESTIMATION D.1 INTRODUCTION

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1 Appendix D INTRODUCTION TO BOOTSTRAP ESTIMATION D.1 INTRODUCTION Bootstrapping is a general, distribution-free method that is used to estimate parameters ofinterest from data collected from studies or experiments. It is often referred to as a resampling method because it is carried out by repeatedly drawing samples from the original data that were gathered. This section introduces the basics of bootstrapping and extends it to bootstrapping in regression analysis. For a discussion on calculating bias or calculating confidence intervals using bootstrapping, see Efron and Tibshirani (1993). Bootstrapping is a useful estimation technique when: 1. The formulas that are to be used for calculating estimates are based on assumptions that may not hold or may not be understood well, or cannot be verified, or are simply dubious. 2. The computational formulas hold only for large samples and are unreliable for small samples or simply not valid for small samples. 3. The computational formulas do not exist. To begin the discussion of bootstrapping techniques, assume that a study or experiment was conducted resulting in a data set x 1,..., x n of size n. This is a trivial case where the data are univariate in nature. Most studies involve collection of data on several variables as in the case of regression analysis studies. However, we use the simple example to lay the groundwork for the elements of bootstrapping methods. Assume that the data set was generated by some underlying distribution f(u). Here, f(u) is the probability density function and may be either continuous or discrete. It may be the case that the true density function is unknown and the functional form of f(u)is, therefore, unknown also. We are interested in estimating the parameter u, which describes some feature of the population from which the data were collected. For instance, u could be the true mean, median, the proportion, the variance, or the standard deviation of the population. Assume for the moment that we have a well-defined formula to calculate an estimate, ^u, of u. However, no formulas exist for calculating the confidence interval for u. Under the ideal setting where we have unlimited resources, we could draw a large number of samples from the population. We could then estimate u by calculating ^u for each sample. The calculated values of ^u can then be used to construct an empirical distribution of ^u that could then be used to construct a confidence interval for u. However, in reality we just have a single sample that is a justification for the use of bootstrapping method. The general idea behind bootstrapping is as follows (assuming that a study/experiment resulted in a data set of size n): 1. A sample of size n is drawn with replacement from the data set in hand. 2. An estimate, ^u, of u is calculated. Applied Econometrics Using the SAS Ò System, by Vivek B. Ajmani Copyright Ó 2009 John Wiley & Sons, Inc. 262

2 APPENDIX D: INTRODUCTION TO BOOTSTRAP ESTIMATION Steps 1 and 2 are repeated several times (sometimes thousands of repetitions are used) to generate a (simulated) distribution of ^u. This simulated distribution is then used for making inferences about u. As an example, suppose that we want to construct a 95% confidence interval for u. However, we do not have formulas that can be used for calculating the interval. We can therefore use bootstrapping to construct the confidence interval. The steps are as follows (Efron and Tibshirani, 1993): 1. Draw 1000 (as an example) bootstrap samples from the original data and calculate ^u 1 ;...; ^u 1000, the estimates from each of the 1000 samples. 2. Next, sort these estimates in increasing order. 3. Calculate the 2.5th and 97.5th percentile from the 1000 simulated values of ^u. The 2.5th percentile will be the average of the 25th and 26th observation while the 97.5th percentile will be the average of the 975th and 976th observation. That is, Lower confidence limit ¼ ^u 25 þ ^u 26 ; 2 Upper confidence limit ¼ ^u 975 þ ^u 976 : 2 Notice that we took the lower 2.5% and the upper 2.5% of the simulated distribution of ^u out to achieve the desired 95% confidence. Also note that we did not make any assumptions about the underlying distribution that generated the original data set. We will now formalize the general bootstrapping method presented so far. Consider a random variable x with cumulative distribution F(x; u). Here, u is a vector of unknown parameters. For example, if the distribution of x is normal, then u ¼ (m, s 2 ). Assume that we are interested in estimating u or some element of u that describes some aspect of f(x; u), the distribution of x. That is, we may be interested in estimating the mean, or the standard deviation, or the standard error of the mean. As we did before, we will assume that a study/experiment resulted in a random sample x 1,..., x n of size n. We can use this sample to approximate the cumulative distribution, F(x; u), with the empirical distribution function, ^Fðx; uþ. The estimate, ^Fðx; uþ, can be written as ^Fðx; uþ ¼ 1 n X n I ð 1;xÞ ðx i Þ; where I is an indicator function that counts the number of x s in the original sample that fall in the interval ( 1, x). This is better illustrated in Figure D.1. In Figure D.1, the true distribution, F(x; u), is given by the smooth line while the estimated function, ^Fðx; uþ, is given by the stepwise representation. The parameter vector u or elements of it could be calculated exactly if the form of F(x; u) were known. FIGURE D.1. Plot comparing actual cumulative versus simulated cumulative distributions. (Graph reproduced with permission from Paul Glewwe, University of Minnesota.)

3 264 APPENDIX D: INTRODUCTION TO BOOTSTRAP ESTIMATION That is,if we knew the exact form of F(x; u),then we could derive the probability density function, f(x; u),or a function t(f)to calculate u. However, assume that the functional form of F(x; u) is unknown and that it was approximated with ^Fðx; uþ. Therefore, one option we have is to replace F(x; u) with ^Fðx; uþ to get the estimated function tð ^FÞ. We can then use tð ^FÞ to calculate an estimate, ^u,ofu. The estimator ^u in this instance is called the plug-in estimator of u (Efron and Tibshirani, 1993, p. 35). As an example, the plug-in estimator of the population mean m x, is the sample mean m x ¼ ð1 1 x ¼ 1 n xf ðxþ dx; X n Notice that calculating the mean of x was trivial and did not require bootstrapping methods. In general, bootstrapping techniques are used to calculate standard errors and for constructing confidence intervals without making any assumption about the underlying distribution from which the samples are drawn. x i : D.2 CALCULATING STANDARD ERRORS We will now discuss how bootstrapping methods can be used to calculate an estimate of the standard error of the parameter of interest. Assume then that we have an estimate of u. That is, ^u was calculated from the original data set without the use of bootstrapping. Bootstrapping, however, will be used to calculate an estimate of the standard error of ^u. The general method for doing this is as follows (again assume that we have a data set of size n) (Efron and Tibshirani, 2004, p. 45): 1. Draw B samples of size n with replacement from the original data set. 2. Calculate ^u for each of the samples from step 1. That is, we now have ^u 1 ;...; ^u B. 3. We calculate the standard error from the B estimates of u by using the standard formulas for standard errors. That is, vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 X B se B ð^uþ ¼t ð^u i ^uþ 2 ; B 1 where ^u PB 1 ¼ B ^u i is simply the mean of the ^u 1 ;...; ^u B. In practice, B is set to a very large number. Most practitioners use bootstrapped samples. D.3 BOOTSTRAPPING IN SAS Bootstrapping can easily be programmed in SAS by using simple routines. SAS macros to calculate bootstrapped estimates are available for download from the SAS Institute. The macros can be used to calculate bootstrapped and jackknife estimates for the standard deviation and standard error, and they are also used to calculate the bootstrapped confidence intervals. The macros can also be used to calculate bootstrapped estimates of coefficients in regression analysis. These macros need to be invoked from within SAS. Wewill illustrate the use of these macros a bit later. For now, we show howasimple program can bewritten to compute bootstrap estimates. Consider a data set that consists of 10 values: 196, 12, 280, 212, 52, 100, 206, 188, 100, 202. We will calculate bootstrap estimates of the standard error for the mean. The following SAS statements can be used: data age_data; input age; cards; 45

4 APPENDIX D: INTRODUCTION TO BOOTSTRAP ESTIMATION ; data bootstrap; do index=1 to 500; do i=1 to nobs; x=round(ranuni(0)*nobs); set age_data nobs=nobs point=x; output; end; end; stop; The following Proc Univariate statements will calculate the mean of the bootstrapped samples. proc univariate data=bootstrap noprint; var age; by index; output out=out1 mean=mean n=n; Finally, the following Proc Univariate statements will calculate the standard deviation of the 500 bootstrapped means. proc univariate data=out1 noprint; var mean; output out=out2 n=n mean=mean std=se; proc print data=out2; The analysis results in a mean and standard error of 27.6 and 6.8, respectively. D.4 BOOTSTRAPPING IN REGRESSION ANALYSIS Consider the standard linear regression model y i ¼ x T i b þ «i, wherex i and b are k 1 columnvectors and e i is randomerror. Assume that we have a data set comprising n pairs of observations (y 1, x 1 ),...,(y n, x n ). Assume that the conditional expectation E(«i jx i ) ¼ 0.Furthermore,assume that we donot know F(«jx),the cumulative distribution ofe.ingeneral,f isassumed tobenormal.

5 266 APPENDIX D: INTRODUCTION TO BOOTSTRAP ESTIMATION We will make use the standard least squares estimator for b, namely, ^b ¼ðX T XÞ 1 X T y, to calculate bootstrapped estimates. That is, as was the case, with the mean being calculated without the use of bootstrapping, we will assume that the least squares estimate can be calculated without any need of bootstrapping. However, we are interested in calculating the standard errors of ^b. That is, we assume that the formulas for calculating the standard errors are either unknown, unreliable, or simply do not work for small samples. As shown in Chapter 1, the estimate of the variance of ^b is Varð^bjXÞ ¼^s 2 ðx T XÞ 1, where ^s 2 is estimated as or ^s 2 ¼ 1 n X n ðy i x T i ^bþ 2 ^s 2 ¼ 1 X n ðy i x T n k 1 ^bþ 2 i : Notice that the first version is not an unbiased estimator for s 2 whereas the second version is. These versions are often referred to as the not bias-corrected and the bias-corrected versions, respectively. There are two bootstrapped methods (pairs method, residuals method) that are employed to estimate the standard error of ^b (Glewwe, 2006; Efron and Tibshirani, 1993, p. 113). The bootstrapped pairs method randomly selects pairs of y i and x i to calculate an estimate of «i, while the bootstrapped residuals method takes each x i just once but then links it with a random draw of an estimate of e. The next section outlines both methods. D.4.1 Bootstrapped Residuals Method As before, we assume that a study or experiment resulted in n observations (y 1, x 1 ),...,(y n, x n ). The general method for the bootstrapped residuals method is 1. For each i, calculate an estimate, e i of «i. That is, e i ¼ y i x T ^b i where ^b is the usual OLS estimator calculated from the original data. 2. Randomly draw n values of e i (from step 1) with replacement. Denote the residuals in the sample as e 1 ; e 2 ;...; e n. Notice that the subscripts of the residuals in the selected sample are not the same as the subscripts for the residuals, e i, which were calculated from the original sample. That is, in general e i 6¼ e i for i ¼ 1,..., n. 3. With the values of e i (from step 2), compute y i ¼ x T ^b i þ e i. Notice that the subscripts for x i here match the subscripts of x i in the original data set. That is, we are using each x i only once. Notice also that by construction of e i, y i 6¼ y i. 4. Using the calculated values of y i (from step 3), construct the vector y *. Finally, use X ¼ x 1... x n Š T and y ¼ y 1... y n ŠT to calculate b 1, the first bootstrapped estimate of b. That is, b 1 ¼ðXT XÞ 1 X T y. 5. Steps 2 through 4 are repeated B (typically B ¼ ) times to get B estimates of b. 6. Use the B estimates (from step 5) to calculate the sample standard deviation of ^b using the formula where vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P u B ð^b s:e: i ^b t Þ 2 ¼ ; B 1 ^b ¼ 1 B is the mean of the B residuals method bootstrapped estimates of b. X B ^b i

6 APPENDIX D: INTRODUCTION TO BOOTSTRAP ESTIMATION 267 D.4.2 Bootstrapped Pairs Method As before, we assume that a study or experiment resulted in n observations (y 1, x 1 ),...,(y n, x n ). The general method for the bootstrapped pairs method is 1. Randomly draw n pairs of values, y i and x i, with replacement. Denote these as y 1 ; y 2 ;...; y n and x 1 ; x 2 ;...; x n.as discussed earlier, the subscripts here do not necessarily match the subscripts in the original data set. 2. Using these values of y i and x i, calculate the first bootstrapped estimate of b by using standard OLS techniques. That is, b 1 ¼ðXT X Þ 1 X T y. 3. Steps 1 and 2 are repeated B times (typically B ¼ ) to get B estimates of b. 4. Use the B estimates b i ; i ¼ 1;...; B, to calculate the sample standard deviation of ^b using the formula where vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P u B ðb s:e: i ^b t Þ 2 ¼ ; B 1 ^b ¼ 1 B X B b i is the mean of the B pairs method bootstrapped estimates of b. Computationally, the bootstrapped pairs method is more straightforward. As discussed in Efron and Tibshirani (1993, p. 113), the bootstrapped residuals method imposes homoscedasticity because it delinks x i with e i. Therefore, if the homoscedasticity assumption is violated, then we should use the bootstrapped pairs method, which does not impose this. On the other hand, if we are very confident of homoscedasticity, then we can use the bootstrapped residuals method to get more precise estimates of the standard error of ^b. In fact, it can be shown that as B!1the standard errors of the least squares estimates calculated using the bootstrapped residuals method converge to the diagonal elements of the variance covariance matrix ^s 2 ðx T XÞ 1. D.4.3 Bootstrapped Regression Analysis in SAS We will now illustrate the residuals and the pairs methods by using the %BOOT macro that can be downloaded from the SAS Institute website at We will make use of the gasoline consumption data given in Table F2.1 of Greene (2003). We need the bootstrapped macros (labeled JACKBOOT.SAS here) to be called from within the program. The %include statement can be used for this purpose. The following statements can be used: %include "C:\Temp\jackboot.sas"; The data set is then read into SAS and stored into a temporary SAS data set called gasoline. Notice that the raw data are stored in Excel format. proc import out=gasoline datafile="c:\temp\gasoline" dbms=excel Replace; getnames=yes; The following SAS data step statements simply transform the variables in the raw data by using the log transformations: data gasoline; set gasoline; Ln_G_Pop=log(G/Pop);

7 268 APPENDIX D: INTRODUCTION TO BOOTSTRAP ESTIMATION Ln_pg=Log(Pg); Ln_Income=Log(Y/Pop); Ln_Pnc=Log(Pnc); Ln_Puc=Log(Puc); The following Proc Reg statements are used to run OLS regression on the original data set. The residuals from this are stored in a temporary SAS data set called gasoline. The residuals are labeled as resid. proc reg data=gasoline; model Ln_G_Pop=Ln_pg Ln_Income Ln_Pnc Ln_Puc; output out=gasoline r=resid p=pred; The following macro is required before invoking the bootstrapped macros in the program jackboot.sas. The only inputs that require changes are the variable names in the model statement. The remaining statements can be used as is. See Sample JackKnife and Bootstrap Analyses from the SAS Institute for more details. The following code has been adapted from this publication and has been used with permission from the SAS Institute. %macro analyze(data=,out=); options nonotes; proc reg data=&data noprint outest=&out(drop=y _IN P EDF_); model Ln_G_Pop=Ln_pg Ln_Income Ln_Pnc Ln_Puc; %bystmt; options notes; %mend; This portion of the code invokes the %boot macro within jackboot.sas and conducts a bootstrapped analysis by using the pairs method. Note that the root mean square error (_RMSE_) is not a plug-in estimator for s, and therefore the bias correction is wrong. In other words, even though the mean square error is unbiased for s 2, the root mean square error is not unbiased for s. However, we choose to ignore this because the bias is minimal. title2 Resampling Observations-Pairs Method ; title3 (Bias correction for _RMSE_ is wrong) ; %boot(data=gasoline, random=123); This portion of the code invokes the %boot macro and conducts the bootstrapped analysis by using the residuals method. title2 Resampling Residuals-Residual Method ; title3 (bias correction for _RMSE_ is wrong) ; %boot(data=gasoline, residual=resid, equation=y=pred+resid, random=123); The analysis results are given in Outputs D.1 and D.2. The first part of the output is from the analysis of the original data. Wewill skip any discussion of this portion of the output as we have already discussed OLS regression output from SAS in detail in Chapter 2. The OLS output is followed by the output where bootstrapping is done by resampling pairs (Output D.1) and where the analysis was done using the residuals method (Output D.2).

8 APPENDIX D: INTRODUCTION TO BOOTSTRAP ESTIMATION 269 Resampling Observations (bias correction for _RMSE_ is wrong) The REG Procedure Model: MODEL1 Dependent Variable: Ln_G_Pop Number of Observations Read 36 Number of Observations Used 36 Analysis of Variance Source DF Sum of Squares Mean Square F Value Pr > F Model < Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var Parameter Estimates Variable DF Parameter Estimate Standard Error t Value Pr > t Intercept < Ln_pg Ln_Income < Ln_Pnc Ln_Puc Resampling Observations (bias correction for _RMSE_ is wrong) Standard Error Lower Limit Name Observed Statistic Bootstrap Mean Bias Bias-Corrected Statistic Intercept Ln_G_Pop Ln_Income Ln_Pnc Ln_Puc Ln_pg _RMSE_ Upper Limit Method for Interval Minimum Maximum Resampled Resampled Estimate Estimate Number of LABEL OF FORMER Name Level (%) Resamples VARIABLE Intercept Bootstrap Normal Intercept Ln_G_Pop Bootstrap Normal Ln_Income Bootstrap Normal Ln_Pnc Bootstrap Normal Ln_Puc Bootstrap Normal Ln_pg Bootstrap Normal _RMSE_ Bootstrap Normal Root mean squared error OUTPUT D.1. Bootstrapped regression analysis (pairs method) of the gasoline consumption data.

9 270 APPENDIX D: INTRODUCTION TO BOOTSTRAP ESTIMATION The REG Procedure Model: MODEL1 Dependent Variable: Ln_G_Pop Number of Observations Read 36 Number of Observations Used 36 Analysis of Variance Source DF Sum of Squares Mean Square F Value Pr > F Model < Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var Parameter Estimates Variable DF Parameter Estimate Standard Error t Value Pr > t Intercept < Ln_pg Ln_Income < Ln_Pnc Ln_Puc Standard Error Lower Limit Name Observed Statistic Bootstrap Mean Bias Bias-Corrected Statistic Intercept E Ln_G_Pop Ln_Income E Ln_Pnc E Ln_Puc E Ln_pg E _RMSE_ E Upper Limit Resampling Residuals (bias correction for _RMSE_ is wrong) Method for Interval Minimum Maximum Resampled Resampled Estimate Estimate Number of LABEL OF FORMER Name Level (%) Resamples VARIABLE Intercept Bootstrap Normal Intercept Ln_G_Pop Bootstrap Normal Ln_Income Bootstrap Normal Ln_Pnc Bootstrap Normal Ln_Puc Bootstrap Normal Ln_pg Bootstrap Normal _RMSE_ Bootstrap Normal Root mean squared error OUTPUT D.2. Bootstrapped regression analysis (residuals method) of the gasoline consumption data.

10 APPENDIX D: INTRODUCTION TO BOOTSTRAP ESTIMATION 271 The output consists of the OLS estimates in the first column, followed by the mean of the coefficients estimated from the 200 bootstrap samples. The third column gives the bias, which is simply the bootstrap mean minus the observed statistic. The standard errors calculated from the bootstrapped samples are given next. This is followed by the 95% confidence intervals, the biascorrected statistics, and the minimum and maximum of the estimated coefficient values from the bootstrap samples. Notice that the bootstrap estimates of the coefficients and the standard errors are very similar to the OLS estimates. There is a remarkable similarity between the bootstrap estimates of the coefficients and the standard errors obtained from the residual method and the OLS estimates. This is not surprising since under the homoscedastic assumption, it can be shown that as the number of bootstrapped samples increases, the estimated values of the standard errors converge to the diagonal elements of ^sðx T XÞ 1, where ^s 2 is the estimate that is not corrected for bias.