ࡱ> _a^q` $bjbjqPqP 8::3%   rrr8N4 4l"22   2222222$6h~82Y=  ==222D4=p222=20|12 DBrp12Z4041(9^(9$1(91 "/G[   22{^   4====   DN$   N   D4 Violations of Classical Linear Regression Assumptions Mis-Specification Assumption 1. Y=Xb+e a. What if the true specification is Y=Xb+Zg+e but we leave out the relevant variable Z? Then the error in the estimated equation is really the sum Zb+e. Multiply the true regression by X to get the mis-specified OLS: X Y=X Xb+X Zg+X e. The OLS estimator is b=(X X)-1X Y= (X X)-1X Xb+(X X)-1X Zg+(X X)-1X e. The last term is on average going to vanish, so we get b=b+(X X)-1X Zg. Unless g=0 or in the data, the regression of X on Z is zero, the OLS b is biased. b. What if the true specification is Y=Xb+e but we include the irrelevant variable Z: Y=Xb+Zg+(e-Zg). The error is e*=e-Zg. Var(e*)=var(e)+g var(Z)g. The estimator of [b g] is  EMBED Equation.3  The expected value of this is  EMBED Equation.3 . Thus the OLS produces an unbiased estimate of the truth when irrelevant variables are added. However, the standard error of the estimate is enlarged in general by gZZg/(n-k) (since e*e*=ee-2eZg+gZZg). This could easily lead to the conclusion that b=0 when in fact it is not. c. What if the coefficients change within the sample, so b is not a constant? Suppose that bi=b+Zig. Then the proper model is Y=X(b+Zg)+e=Xb+XZg+e. Thus we need to include the interaction term XZ. If we do not, then we are in the situation (a) above, and the OLS estimates of the coefficients of X will be biased. On the other hand, if we include the interaction term when it is not really appropriate, the estimators are unbiased but not minimum variance. We can get fooled about the true value of b. How do you test whether the interactions belong or not. Run an unconstrained regression (which includes interactions) and then run a constrained regression (set interaction coefficients equal to zero). [(SSEconst-SSEunconst)/q]/[SSEunconst/(n-k)]~ Fq,n-k where q=number of interaction terms. d. Many researchers do a search for the proper specification. This can lead to spurious results and we will look at this is some detail in a lecture to follow. Censored Data and Frontier Regression Assumption 2. E[e|X]=0. Suppose that E[ei |X]=m`"0. Note: this is the same for all i. b=(X X)-1X Y=(X X)-1X (Xb+e) =b+(X X)-1X e. Thus E[b]=b+m(X X)-1X 1. The term (X X)-1X 1 is the regression of 1 on X, but the first column of X is 1 so the resulting regression coefficients must be [1 0 0& 0] . As a result E[b]=b+[m 0 0 & 0] . Only the intercept is biased. Now suppose that E[ei|X]=mi but this varies with i. That is, m`"m1. By reasoning like the above, E[b]=b+(X X)-1X m The regression of m on X will in general have non-zero coefficients everywhere and the estimate of b will be biased in all ways. In particular, what if the data was censored in the sense that only observations of Y that are not too small nor too large are included in the sample: MIN (Yi(MAX. Hence for values of Xi such that Xib are very small or vary large, only errors that are high and low respectively will lead to observations in the dataset. This can lead to the type of bias discussed above for all the coefficients, not just the intercept. See the graph below where the slope is also biased.  Frontier Regression: Stochastic Frontier Analysis Cost Regression: Ci=a + bQi + ei + fi The term a+bQ+e represents the minimum cost measured with a slight measurement error e. Given this, the actual costs must be above the minimum so the inefficiency term f must be positive. Suppose that f has an exponential distribution: f(f)=e-f/l/l for f(0. [Note: E[f]=l and Var[f]=l2.] Suppose that the measurement error e~N(0,s2) and is independent of the inefficiency f. The joint probability of e and f is  EMBED Equation.3 . Let the total error be denoted q=e+f. [Note: E[q]=l and Var[q]=s2+l2.] Then the joint probability of the inefficiency and total error is  EMBED Equation.3 . The marginal distribution of the total error is found by integrating the f(q,f) with respect to f over the range [0,(). Using  complete-the-square this can be seen to equal  EMBED Equation.3 , where F is the cumulative standard normal. To fit the model to n data-points, we would select a, b , l and s to maximize log-likelihood:  EMBED Equation.3  Once we have estimated the parameters, we can measure the amount of inefficiency for each observation, fi. The conditional pdf f(fi|qi) is computed for qi=Ci-a-bQi:  EMBED Equation.3 . This is a half-normal distribution and has a mode of qi-s2/l, assuming this is positive. The degree of cost inefficiency is defined as IEi= EMBED Equation.3 ; this is a number greater than 1, and the bigger it is the more inefficiently large is the cost. Of course, we do not know fi, but if we evaluate IEi at the posterior mode qi-s2/l it equals IEi ( EMBED Equation.3 . Note that the term s2/l captures the idea that we do not precisely know what the minimum cost equals, so we slightly discount the measured cost to account for our uncertainty about the frontier. Non-Spherical Errors Assumption 3. var(Y|X)=var(e|X)=s2 I Suppose that var(e|X)= s2 W, where W is a symmetric, positive definite matrix but W`"I. What are the consequences for OLS? a. E[b]=E[(X X)-1X (Xb+e)]=b+(X X)-1X E[e] = b, so OLS is still unbiased even if W`"I. b. Var[b]=E[(b-b)(b-b) ]=(X X)-1X E[ee ]X(X X)-1=s2(X X)-1X WX(X X)-1`"s2(X X)-1 Hence, the OLS computed standard errors and t-stats are wrong. The OLS estimator will not be BLUE. Generalized Least-Squares Suppose we find a matrix P (n(n) such that PWP =I, or equivalently W=P-1P -1 or W-1=P P (use spectral demcomposition). Multiply the regression model (Y=Xb+e) on left by P: PY=PXb+Pe. Write PY=Y*, PX=X* and Pe=e*, so in the transformed variables Y*=X*b+e*. Why do this? Look at the variance of e*: Var(e*)=E[e*e* ]=E[Pee P ]=PE[ee ]P =s2PWP =s2 I. The error e* is spherical; that s why. GLS estimator: b*=(X* X*)-1X* Y*=(X P PX)-1X P PY=(X W-1X)-1X W-1Y. Analysis of the transformed data equation says that GLS b* is BLUE. So it has lower variance that the OLS b. Var[b*]=s2(X* X*)-1= s2(X W-1X)-1 How do we estimate s2? [Note: from OLS E[e e]/(n-k)=E[e Me]/(n-k)=E[tr(e Me)]/(n-k)=E[tr(Mee )]/(n-k) =tr(ME[ee ])/(n-k)=s2tr(MW)/(n-k). Since W`"I, tr(MW)`"n-k, so E[e e]/(n-k) `"s2.] Hence, to estimate s2 we need to use the errors from the transformed equation Y*=X*b*+e*. s*2=(e* e*)/(n-k) E[s*2]=tr(M*E[e*e* ])/(n-k)= s2tr(M*PWP )/(n-k)= s2tr(M*)/(n-k)=s2. Hence s*2 is an unbiased estimator of s2. Important Note: all of the above assumes that W is known and that it can be factored into P-1P -1. How do we know W? Two special cases are autocorrelation and heteroskedasticity. Autocorrelated Errors Suppose that Yt=Xtb+ut (notice the subscript t denotes time since this problem occurs most frequently with time-series data). 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Successively lagging and substituting for ut gives the equivalent formula ut=et+ret-1+r2et-2+& Using this, we can see that E[utut]=s2(1+r2+r4+& )=s2/(1-r2), E[utut-1]=r s2/(1-r2), E[utut-2]=r2 s2/(1-r2), & E[utut-m]=rm s2/(1-r2). Therefore, the variance matrix of u is var(u)=E[uu ] = EMBED Equation.3 =s2W, where  EMBED Equation.3  and  EMBED Equation.3  It is possible to show that W-1 can be factored into P P where  EMBED Equation.3 . Given this P, the transformed data for GLS is  EMBED Equation.3  Notice that only the first element is unique. The rest just involves subtracting a fraction r of the lagged value from the current value. Many modelers drop the first observation and use only the last n-1 because it is easier, but this throws away information and I would not recommend doing it unless you had a very large n. The Cochrane-Orcutt technique successively estimates of r from the errors and re-estimating based upon new transformed data (Y*,X*). 1. Guess a starting r0. 2. At stage m, estimate b in model Yt-rmYt-1=(Xt-rmXt-1)b+et using OLS. If the estimate bm is not different from the previous bm-1, then stop. Otherwise, compute error vector em=(Y*-X*bm). 3. Estimate r in emt=rem,t-1+et via OLS. This estimate becomes the new rm+1. Go back to 2. Durbin-Watson test for r`"0 in ut=rut-1+et. 1. Compute OLS errors e. 2. Calculate  EMBED Equation.3 . 3. d<2 ( r>0, d>2 ( r<0, d=2 ( r=0.  Heteroskedasticity Here we assume that the errors are independent, but not necessarily identically distributed. That is the matrix W is diagonal, but not the identity matrix. The most common way for this to occur is because Yi is the average response of a group i that has a number of members mi. Larger groups have smaller variance in the average response: var(ei)=s2/mi. Hence the variance matrix would be Var(e)= EMBED Equation.3 . An related example of this would be that Y is the sum across the members of many similar elements, so that the var(ei)=s2 mi and Var(e)= EMBED Equation.3 . If we knew how big the groups where and whether we had the average or total response, we could substitute for mi in the above matrix W. More generally, we think that the variance of eI depends upon some variable Z. We can do a Glessjer Test of this as follows. 1. Compute OLS estimate of b,e 2. Regress |ei| on Zih, where h=1,-1, and . 3. If the coefficient of Zh is 0 then the model is homoscedastic, but if it is not zero, then the model has heteroskedastic errors. In SPSS, you can correct for heteroskedasticity by using Analyze/Regression/Weight Estimation rather than Analyze/Regression/Linear. You have to know the variable Z, of course. Trick: Suppose that st2=s2Zt2. Notice Z is squared. Divide both sides of equation by Z to get Yt/Zt=(Xt/Zt)b+et/Zt. This new equation has homoscedastic errors and so the OLS estimate of this transformed model is BLUE. Simultaneous Equations Assumption 4. X is fixed Later in the semester will return to the problem that X is often determined by actors in the play we are studying rather than by us scientists. This is a serious problem in simultaneous equation models. Multicollinearity Assumption 5. X has full column rank. What is the problem if you have multicollinearity? In X X there will be some portions that look like a little square  EMBED Equation.3  and this has a determinant equal to zero, so its reciprocal will be near infinity. OLS is still BLUE, but estimated var[b]=(X X)-1Y (I-X(X X)-1X )Y/(n-k) can be very large. If there is collinearity, then there exists a weighting vector a such that Xa is close to the 0 vector. Of course, we cannot just allow a to be zero. Hence let s look for the value of a that minimizes ||Xa||2 subject to a a=1. The Lagrangian for this constrained optimization is L=a X Xa+l(1-a a) and the first order conditions are X Xa-la=0 This is the equation for the eigenvalue and eigenvector of X X. Multiply the first order condition by a and use the fact that eigenvectors have a length of 1 to see that a X Xa=l, so we are looking at the smallest of the eigenvalues when we seek collinearity. When is this eigenvalue  small enough to measure serious collinearity? We compute a Condition Index as the square root of the ratio largest eigenvalue to the smallest eigenvalue: EMBED Equation.3 . When the condition index is greater than 20 or 30, we have serious collinearity. In SPSS Regression/Linear/Statistics click  Collinearity Diagnostics. Warning: Many people use the Variance Inflation Factor to identify collinearity. This should be avoided (see Chennamaneni, Echambadi, Hess and Syam 2009). The problem is that VIF confuses  collinearity with  correlation as follows. Let R be the correlation matrix of X: R=D-X HXD-/(n-1) where the standard deviation matrix D=sqrt(diag(X HX)/(n-1)). Compute R-1. For example,  EMBED Equation.3  and along the diagonal is 1/(1-r2) which is called the Variance Inflation Factor (VIF). More generally VIFi=(1-Ri2)-1 where Ri2 is the R-square from regressing xi on the k-1 other variables in X. The problem with VIF is that it starts with a mean-centered data HX, when collinearity is a problem of the raw data X. In OLS we compute (X X)-1, not (X HX)-1. Chennamani et al. provide a variant of VIF that does not suffer from these problems. What can you do if there is collinearity? 1) Do nothing. OLS is BLUE. 2) Get more information. Obtain more data or formalize the links between the elements of X. 3) Summarize X. Drop a variable or do principal component analysis (more on this in next chapter of the textbook). 4) Use ridge regression. This appends a matrix kI to the bottom of the exogenous data X and appends a corresponding vector of 0 s to the bottom of the endogenous data Y. This synthetic data obviously results in a biased estimator (biased toward 0 since the augmented data has Y not responding to changes in X), but the augmented data kI has orthogonal and hence maximally  not collinear observations. Hence, the estimates become more precise. For k(0, the improved precision dominates the bias.  Aigner, D., C. Lovell and P. Schmidt (1977),  Specification and Estimation of Production Frontier Production Function Models, J. Econometrics, 6:1 (July), 21-37; Kumbhaka, S and C. Lovell (2000), Stochastic Frontier Analysis, Cambridge Univ Press. Free SFA software FRONTIER 4.1 is available at  HYPERLINK "http://www.uq.edu.au/economics/cepa/frontier.htm" http://www.uq.edu.au/economics/cepa/frontier.htm .     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