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Part 7: Estimating the Variance of b 7-1/53 Econometrics I Professor William Greene Stern School of Business Department of Economics

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Part 7: Estimating the Variance of b 7-2/53 Econometrics I Part 7 – Estimating the Variance of b

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Part 7: Estimating the Variance of b 7-3/53 Context The true variance of b|X is 2 (XX) -1. We consider how to use the sample data to estimate this matrix. The ultimate objectives are to form interval estimates for regression slopes and to test hypotheses about them. Both require estimates of the variability of the distribution. We then examine a factor which affects how "large" this variance is, multicollinearity.

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Part 7: Estimating the Variance of b 7-4/53 Estimating 2 Using the residuals instead of the disturbances: The natural estimator: ee/N as a sample surrogate for /n Imperfect observation of i, e i = i - ( - b)x i Downward bias of ee/N. We obtain the result E[ee|X] = (N-K) 2

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Part 7: Estimating the Variance of b 7-5/53 Expectation of ee

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Part 7: Estimating the Variance of b 7-6/53 Method 1:

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Part 7: Estimating the Variance of b 7-7/53 Estimating σ 2 The unbiased estimator is s 2 = ee/(N-K). “Degrees of freedom correction” Therefore, the unbiased estimator of 2 is s 2 = ee/(N-K)

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Part 7: Estimating the Variance of b 7-8/53 Method 2: Some Matrix Algebra

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Part 7: Estimating the Variance of b 7-9/53 Decomposing M

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Part 7: Estimating the Variance of b 7-10/53 Example: Characteristic Roots of a Correlation Matrix

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Part 7: Estimating the Variance of b 7-11/53

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Part 7: Estimating the Variance of b 7-12/53 Gasoline Data

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Part 7: Estimating the Variance of b 7-13/53 X’X and its Roots

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Part 7: Estimating the Variance of b 7-14/53 Var[b|X] Estimating the Covariance Matrix for b|X The true covariance matrix is 2 (X’X) -1 The natural estimator is s 2 (X’X) -1 “Standard errors” of the individual coefficients are the square roots of the diagonal elements.

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Part 7: Estimating the Variance of b 7-15/53 X’X (X’X) -1 s 2 (X’X) -1

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Part 7: Estimating the Variance of b 7-16/53 Standard Regression Results ---------------------------------------------------------------------- Ordinary least squares regression........ LHS=G Mean = 226.09444 Standard deviation = 50.59182 Number of observs. = 36 Model size Parameters = 7 Degrees of freedom = 29 Residuals Sum of squares = 778.70227 Standard error of e = 5.18187 <= sqr[778.70227/(36 – 7)] Fit R-squared =.99131 Adjusted R-squared =.98951 --------+------------------------------------------------------------- Variable| Coefficient Standard Error t-ratio P[|T|>t] Mean of X --------+------------------------------------------------------------- Constant| -7.73975 49.95915 -.155.8780 PG| -15.3008*** 2.42171 -6.318.0000 2.31661 Y|.02365***.00779 3.037.0050 9232.86 TREND| 4.14359** 1.91513 2.164.0389 17.5000 PNC| 15.4387 15.21899 1.014.3188 1.67078 PUC| -5.63438 5.02666 -1.121.2715 2.34364 PPT| -12.4378** 5.20697 -2.389.0236 2.74486 --------+-------------------------------------------------------------

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Part 7: Estimating the Variance of b 7-17/53 Bootstrapping Some assumptions that underlie it - the sampling mechanism Method: 1. Estimate using full sample: --> b 2. Repeat R times: Draw N observations from the n, with replacement Estimate with b(r). 3. Estimate variance with V = (1/R) r [b(r) - b][b(r) - b]’

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Part 7: Estimating the Variance of b 7-18/53 Bootstrap Application matr;bboot=init(3,21,0.)$ Store results here name;x=one,y,pg$ Define X regr;lhs=g;rhs=x$ Compute b calc;i=0$ Counter Proc Define procedure regr;lhs=g;rhs=x;quietly$ … Regression matr;{i=i+1};bboot(*,i)=b$... Store b(r) Endproc Ends procedure exec;n=20;bootstrap=b$ 20 bootstrap reps matr;list;bboot' $ Display results

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Part 7: Estimating the Variance of b 7-19/53 --------+------------------------------------------------------------- Variable| Coefficient Standard Error t-ratio P[|T|>t] Mean of X --------+------------------------------------------------------------- Constant| -79.7535*** 8.67255 -9.196.0000 Y|.03692***.00132 28.022.0000 9232.86 PG| -15.1224*** 1.88034 -8.042.0000 2.31661 --------+------------------------------------------------------------- Completed 20 bootstrap iterations. ---------------------------------------------------------------------- Results of bootstrap estimation of model. Model has been reestimated 20 times. Means shown below are the means of the bootstrap estimates. Coefficients shown below are the original estimates based on the full sample. bootstrap samples have 36 observations. --------+------------------------------------------------------------- Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------- B001| -79.7535*** 8.35512 -9.545.0000 -79.5329 B002|.03692***.00133 27.773.0000.03682 B003| -15.1224*** 2.03503 -7.431.0000 -14.7654 --------+------------------------------------------------------------- Results of Bootstrap Procedure

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Part 7: Estimating the Variance of b 7-20/53 Bootstrap Replications Full sample result Bootstrapped sample results

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Part 7: Estimating the Variance of b 7-21/53 OLS vs. Least Absolute Deviations ---------------------------------------------------------------------- Least absolute deviations estimator............... Residuals Sum of squares = 1537.58603 Standard error of e = 6.82594 Fit R-squared =.98284 --------+------------------------------------------------------------- Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------- |Covariance matrix based on 50 replications. Constant| -84.0258*** 16.08614 -5.223.0000 Y|.03784***.00271 13.952.0000 9232.86 PG| -17.0990*** 4.37160 -3.911.0001 2.31661 --------+------------------------------------------------------------- Ordinary least squares regression............ Residuals Sum of squares = 1472.79834 Standard error of e = 6.68059 Standard errors are based on Fit R-squared =.98356 50 bootstrap replications --------+------------------------------------------------------------- Variable| Coefficient Standard Error t-ratio P[|T|>t] Mean of X --------+------------------------------------------------------------- Constant| -79.7535*** 8.67255 -9.196.0000 Y|.03692***.00132 28.022.0000 9232.86 PG| -15.1224*** 1.88034 -8.042.0000 2.31661 --------+-------------------------------------------------------------

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Part 7: Estimating the Variance of b 7-22/53 Quantile Regression: Application of Bootstrap Estimation

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Part 7: Estimating the Variance of b 7-23/53 Quantile Regression Q(y|x, ) = x, = quantile Estimated by linear programming Q(y|x,.50) = x,.50 median regression Median regression estimated by LAD (estimates same parameters as mean regression if symmetric conditional distribution) Why use quantile (median) regression? Semiparametric Robust to some extensions (heteroscedasticity?) Complete characterization of conditional distribution

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Part 7: Estimating the Variance of b 7-24/53 Estimated Variance for Quantile Regression Asymptotic Theory Bootstrap – an ideal application

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Part 7: Estimating the Variance of b 7-25/53

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Part 7: Estimating the Variance of b 7-26/53 =.25 =.50 =.75

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Part 7: Estimating the Variance of b 7-27/53

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Part 7: Estimating the Variance of b 7-28/53

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Part 7: Estimating the Variance of b 7-29/53 Multicollinearity Not “short rank,” which is a deficiency in the model. A characteristic of the data set which affects the covariance matrix. Regardless, is unbiased. Consider one of the unbiased coefficient estimators of k. E[b k ] = k Var[b] = 2 (X’X) -1. The variance of b k is the kth diagonal element of 2 (X’X) -1. We can isolate this with the result in your text. Let [X,z] be [Other xs, x k ] = [X 1,x 2 ] (a convenient notation for the results in the text). We need the residual maker, M X. The general result is that the diagonal element we seek is [zM 1 z] -1, which we know is the reciprocal of the sum of squared residuals in the regression of z on X.

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Part 7: Estimating the Variance of b 7-30/53 I have a sample of 24025 observations in a logit model. Two predictors are highly collinear (pairwaise corr.96; p<.001); vif are about 12 for eachof them; average vif is 2.63; condition number is 10.26; determinant of correlation matrix is 0.0211; the two lowest eigen vales are 0.0792 and 0.0427. Centering/standardizing variables does not change the story. Note: most obs are zeros for these two variables; I only have approx 600 non-zero obs for these two variables on a total of 24.025 obs. Both variable coefficients are significant and must be included in the model (as per specification). -- Do I have a problem of multicollinearity?? -- Does the large sample size attenuate this concern, even if I have a correlation of.96? -- What could I look at to ascertain that the consequences of multi-collinearity are not a problem? -- Is there any reference I might cite, to say that given the sample size, it is not a problem? I hope you might help, because I am really in trouble!!!

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Part 7: Estimating the Variance of b 7-31/53 Variance of Least Squares

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Part 7: Estimating the Variance of b 7-32/53 Multicollinearity

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Part 7: Estimating the Variance of b 7-33/53 Gasoline Market Regression Analysis: logG versus logIncome, logPG The regression equation is logG = - 0.468 + 0.966 logIncome - 0.169 logPG Predictor Coef SE Coef T P Constant -0.46772 0.08649 -5.41 0.000 logIncome 0.96595 0.07529 12.83 0.000 logPG -0.16949 0.03865 -4.38 0.000 S = 0.0614287 R-Sq = 93.6% R-Sq(adj) = 93.4% Analysis of Variance Source DF SS MS F P Regression 2 2.7237 1.3618 360.90 0.000 Residual Error 49 0.1849 0.0038 Total 51 2.9086

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Part 7: Estimating the Variance of b 7-34/53 Gasoline Market Regression Analysis: logG versus logIncome, logPG,... The regression equation is logG = - 0.558 + 1.29 logIncome - 0.0280 logPG - 0.156 logPNC + 0.029 logPUC - 0.183 logPPT Predictor Coef SE Coef T P Constant -0.5579 0.5808 -0.96 0.342 logIncome 1.2861 0.1457 8.83 0.000 logPG -0.02797 0.04338 -0.64 0.522 logPNC -0.1558 0.2100 -0.74 0.462 logPUC 0.0285 0.1020 0.28 0.781 logPPT -0.1828 0.1191 -1.54 0.132 S = 0.0499953 R-Sq = 96.0% R-Sq(adj) = 95.6% Analysis of Variance Source DF SS MS F P Regression 5 2.79360 0.55872 223.53 0.000 Residual Error 46 0.11498 0.00250 Total 51 2.90858 The standard error on logIncome doubles when the three variables are added to the equation.

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Part 7: Estimating the Variance of b 7-35/53 Condition Number and Variance Inflation Factors Condition number larger than 30 is ‘large.’ What does this mean?

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Part 7: Estimating the Variance of b 7-36/53

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Part 7: Estimating the Variance of b 7-37/53 The Longley Data

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Part 7: Estimating the Variance of b 7-38/53 NIST Longley Solution

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Part 7: Estimating the Variance of b 7-39/53 Excel Longley Solution

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Part 7: Estimating the Variance of b 7-40/53 The NIST Filipelli Problem

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Part 7: Estimating the Variance of b 7-41/53 Certified Filipelli Results

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Part 7: Estimating the Variance of b 7-42/53 Minitab Filipelli Results

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Part 7: Estimating the Variance of b 7-43/53 Stata Filipelli Results

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Part 7: Estimating the Variance of b 7-44/53 Even after dropping two (random columns), results are only correct to 1 or 2 digits.

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Part 7: Estimating the Variance of b 7-45/53 Regression of x2 on all other variables

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Part 7: Estimating the Variance of b 7-46/53 Using QR Decomposition

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Part 7: Estimating the Variance of b 7-47/53 Multicollinearity There is no “cure” for collinearity. Estimating something else is not helpful (principal components, for example). There are “measures” of multicollinearity, such as the condition number of X and the variance inflation factor. Best approach: Be cognizant of it. Understand its implications for estimation. What is better: Include a variable that causes collinearity, or drop the variable and suffer from a biased estimator? Mean squared error would be the basis for comparison. Some generalities. Assuming X has full rank, regardless of the condition, b is still unbiased Gauss-Markov still holds

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Part 7: Estimating the Variance of b 7-48/53 Specification and Functional Form: Nonlinearity

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Part 7: Estimating the Variance of b 7-49/53 Log Income Equation ---------------------------------------------------------------------- Ordinary least squares regression............ LHS=LOGY Mean = -1.15746 Estimated Cov[b1,b2] Standard deviation =.49149 Number of observs. = 27322 Model size Parameters = 7 Degrees of freedom = 27315 Residuals Sum of squares = 5462.03686 Standard error of e =.44717 Fit R-squared =.17237 --------+------------------------------------------------------------- Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------- AGE|.06225***.00213 29.189.0000 43.5272 AGESQ| -.00074***.242482D-04 -30.576.0000 2022.99 Constant| -3.19130***.04567 -69.884.0000 MARRIED|.32153***.00703 45.767.0000.75869 HHKIDS| -.11134***.00655 -17.002.0000.40272 FEMALE| -.00491.00552 -.889.3739.47881 EDUC|.05542***.00120 46.050.0000 11.3202 --------+------------------------------------------------------------- Average Age = 43.5272. Estimated Partial effect =.066225 – 2(.00074)43.5272 =.00018. Estimated Variance 4.54799e-6 + 4(43.5272) 2 (5.87973e-10) + 4(43.5272)(-5.1285e-8) = 7.4755086e-08. Estimated standard error =.00027341.

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Part 7: Estimating the Variance of b 7-50/53 Specification and Functional Form: Interaction Effect

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Part 7: Estimating the Variance of b 7-51/53 Interaction Effect ---------------------------------------------------------------------- Ordinary least squares regression............ LHS=LOGY Mean = -1.15746 Standard deviation =.49149 Number of observs. = 27322 Model size Parameters = 4 Degrees of freedom = 27318 Residuals Sum of squares = 6540.45988 Standard error of e =.48931 Fit R-squared =.00896 Adjusted R-squared =.00885 Model test F[ 3, 27318] (prob) = 82.4(.0000) --------+------------------------------------------------------------- Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X --------+------------------------------------------------------------- Constant| -1.22592***.01605 -76.376.0000 AGE|.00227***.00036 6.240.0000 43.5272 FEMALE|.21239***.02363 8.987.0000.47881 AGE_FEM| -.00620***.00052 -11.819.0000 21.2960 --------+------------------------------------------------------------- Do women earn more than men (in this sample?) The +.21239 coefficient on FEMALE would suggest so. But, the female “difference” is +.21239 -.00620*Age. At average Age, the effect is.21239 -.00620(43.5272) = -.05748.

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Part 7: Estimating the Variance of b 7-52/53

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Part 7: Estimating the Variance of b 7-53/53

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Part 18: Regression Modeling 18-1/44 Statistics and Data Analysis Professor William Greene Stern School of Business IOMS Department Department of Economics.

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