The covariance is the two variable analog to the variance. The formula for the covariance between two variables is


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1 Regresson Lectures So far we have talked only about statstcs that descrbe one varable. What we are gong to be dscussng for much of the remander of the course s relatonshps between two or more varables. To do ths there are several concepts that t would be good to have a gasp of Covarance and Correlaton Recall that Var( ) = σ = E( µ ) whch we estmate wth the sample varance ( ) S = n The covarance s the two varable analog to the varance. The formula for the covarance between two varables s Cov( X, Y ) = σ = E( µ )( y µ ) y y whch we estmate wth the sample covarance ( )( y y) Sy = n When two varables are negatvely related the covarance wll be negatve. When two varables are postvely related the covarance wll be postve. Go through some scatter plots. One potental problem wth the covarance s that we do not have any dea how bg t was. Wth the varance we could fnd the standard devaton. If we knew the was appromately normal ths standard devaton conveyed useful nformaton. We cannot do ths for the covarance. Another problem wth the covarance s that t depends on unts of measurement. If we are nterested n the relatonshp between ncome and years educaton we would get a dfferent covarance f we defned ncome n thousands of dollars than we would defnng ncome n dollars.
2 To correct these problems we can use correlaton between and y. The correlaton s gven by ρ y σ y = σ σ y whch we estmate wth the sample correlaton ρ y Sy ( )( y y) = = SS y ( ) ( y y) Ths s often called the Person Product Moment Correlaton Coeffcent. Both the correlaton and the sample correlaton must be between and. If two varables are perfectly negatvely correlated then the correlaton wll be. If they are perfectly postvely correlated t wll be. The covarance and correlaton descrbe the strength of the lnear relatonshp between two varables. Nether measure, however, descrbes ths relatonshp. To descrbe the lnear relatonshp between two varables we need smple lnear regresson analyss. Smple Lnear Regresson Analyss What s gong n the populaton? There are two varables that are related. y s the dependent varable. s the ndependent varable. In theory y should be caused by (or dependent on). Suppose that there s a lnear relatonshp between and y such that y = + + e. The way to thnk about ths s that n the populaton y s beng generated as the sum of (an ntercept term), (a functon of the ndependent varable), and e (an error term).
3 We can thnk of and as populaton parameters the same way we thought about µ as a populaton parameter. Note that and E y ( ) = + E( ) = + µ E( y ) = + We have already dealt wth a specal case of ths model. y = + e In ths case E y µ ( ) = = y and E y = = µ. ( ) y Suppose that we had a sample that looked somethng lke the followng y y.. y.... n n Then we would be able to estmate the populaton parameters and. Lets consder the data that you had last week for a homework assgnment. Ths was the data on unemployment duraton and age that you used n the case problem. The sample covarance between these two varables s 76.6 and the correlaton coeffcent s.66. So we know (from lookng at the scatter plot) and from these measures that there s a postve relatonshp between weeks unemployed and age. What sorts of thngs would lead to ths type of postve assocaton? Whch of these varables would be the ndependent varable and whch would be the dependent varable f we were settng ths up as a smple lnear regresson?
4 To ft a lne to ths data would bascally have to pck values for (or estmate) and. The estmate of the parameter wll gve us our yntercept and the estmate of the parameter wll gve us the slope? How can I go about fndng estmates for and? Scatterplot of Weeks Unemployed and Age Weeks Unemployed S r S y y = 76.6 = 65.9 = 4.69 = Age Scatterplot of Weeks Unemployed and Age Weeks Unemployed Age
5 If we have sample data we can wrte our orgnal equaton as e y e y.. =.... e y n n We want these errors to be as small as possble n absolute value. The way that we make them as small as possble n absolute value s to fnd and that mnmze n = ( ) y wth respect to and. When we do ths we would fnd that So Sy 76.6 = = =.537 S 4.69 = y = = 4. WU = Age
6 A Few Specal Propertes of Least Squares Estmates ) The least squares estmate always goes through the means. Ths means that (, y ) s always on the least squares lne. Algebracally, t means that y = + Graphcally t means that Scatterplot of Weeks Unemployed and Age 46.6 Weeks Unemployed Age ) e = ( y ) =. Ths s just a just a product of how we chose the and. It s a mathematcal fact. I cannot really gve you any ntuton for t wthout confusng you.
7 The Coeffcent of Determnaton We mght be nterested n how precse our estmate s. In partcular we mght be nterested n determnng how much better our lne ft than the sample mean of y. We could estmate y wth ts sample mean (one parameter) or we could ft a lne (by estmatng parameters, ). How much better do we do n terms of ft by estmatng two parameters? Note that ( ) ( ) y y = + + e + = + e Usng the above we can wrte ( y y) = ( ) + e + ( ) e = ( ) + e where SST = SSR + SSE SST = ( y y ) SSR = ( ) SSE = e The coeffcent of determnaton ( Rsquared ), defned as R SSR SSE = = SST SST s a measure of goodness of ft. It bascally tells us how much better than lne fts than the sample mean of y. Another way to thnk about Rsquared s that t tells us the percentage of the varaton n y away from ts mean s eplaned by versus unknown factors (the resduals). It s useful to thnk of a couple specfc cases.
8 Scatterplot of Weeks Unemployed and Age 46.6 Weeks Unemployed Age Ok Lets thnk about the case when and y are not correlated. The mean of y does just as good as any lne. Indeed the slope estmate should be close to zero so the mean of y s close to the least squares lne. One thng you should know for smple lnear regresson analyss (wth one ndependent varable) the square root of Rsquared s the sample correlaton.
9 Inference and Smple Lnear Regresson Thus far we have talked only about the mechancs of least squares. We have not talked about how to use estmates to make nferences about populaton parameters. Ths s the net step. Before we do ths we need to make some assumptons Assumptons ) We got the functonal form correct: y = + + e ) Zero mean dsturbance: Ee ( ) = 3) Constant varance dsturbance: Var( e ) 4) Errors not autocorrelated: Cov( e, e ) = j 5) Regressors are nonstochastc: Var( ) = j = σ  Constant var 6) e N(, σ )  Ths s not strctly necessary but does allow us to make nferences n the case when sample szes are not very large. The Samplng Dstrbuton of If we want to fnd the samplng dstrbuton of we need to specfy the mean, the varance, and the dstrbuton of. ) Lets workng on fndng the mean. ( )( y y) ( )( ( ) + e) = = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) e e = + = + Takng the epected value we fnd ( ) e E( ) E = + = ( ) So has mean. It s an unbased estmator. Equvalent to E ( ) = µ
10 ) Lets fnd the varance ( ) ( ) e Var( ) = E = E ( ) = E ( ) e = ( ) ( ) = = ( ) Equvalent to σ σ ( ) Var( ) = σ n 3) What s the dstrbuton of? If the dsturbances are normal then s lnear n the dsturbances so s normal If the dsturbances are not normal and the sample sze s large enough we have a central lmt theorem that says the dstrbuton of s appromately normal. Ok so all that we need to do s fnd an estmate of σ and we are n busness. Remember that σ =Var( e ). Thus the approprate estmator s S e SSE = = n n Now we can make nferences about.
11 Confdence Intervals and Hypothess Testng From our estmate wth the BLS data Note that.537 = 4.6 S = 56.9 Now we can estmate Var( ) S S σ σ 56.9 = = = ( ) 56.9 = = ( n ) S 669 Eample) Lets use ths to test the hypothess H H : = a : In words  we are testng the null that age as no effect on the duraton of unemployment spells aganst the alternatve that t does. Choose level of sgnfcance (. ) Get the value of the test statstc.537 z = =5.97 we would reject the null at almost any sgnfcance level..9 Eample) We could also construct a 95% confdence nterval.
12 Multple Regresson Here the model s smlar to the smple lnear model we already talked about. The only dfference beng that y s a lnear functon of a bunch more than one varable not just. That s y = kk + e For ths model E( y,..., k ) = kk E( y) = + µ + µ µ k 3 Var( y ) = Var( e ) = σ Instead of just estmatng 3 populaton parameters Var( e ) = σ we wll be estmatng k+ parameters through k  k+ parameters Var( e ) = σ We wll use least squares to ft our lne (actually t s a plane of dmenson k n k+ dmensonal space). Ths means that through k are the partcular s that mnmze ( y... ) k k Unlke the case of the smple lnear regresson model we are gong to be unable to wrte down formulas for through k. As you shall see we wll stll be able to wrte out a formula for.
13 Goodness of ft and Coeffcent of Determnaton Almost everythng that I sad about the Rsquared n the bvarate regresson model holds for the multvarate model. That s where R SSR SSE = = SST SST SST = ( y y) SSR = ( y y) SSE = e Because you can always mprove Rsquared by addng more varables to the model there s somethng called the adjusted Rsquared n Ra = ( R ) n k I do not epect you to remember ths, but you should understand the purpose for the adjusted Rsquared. Propertes of Multvarate Least Squares ) The least squares lne goes through the means y = k k Here s were the formula for comes n = y... k k ) The least squares resduals wpe out the s e = = j j,..., k
14 Assumptons ) We got the functonal form correct: y = e ) Zero mean dsturbance: Ee ( ) = 3) Constant varance dsturbance: Var( e ) 4) Errors not autocorrelated: Cov( e, e ) = j j k k = σ  Constant varance 5) Regressors are nonstochastc: Var( ) = =,..., n and j =,..., k 6) e N(, σ )  Ths s not strctly necessary but does allow us to make nferences n the case when sample szes are not very large. j Statstcal Inference and the Lnear Regresson Model We are gong to want to characterze the dstrbuton of,..., k. To do ths we need three thngs () The mean of our estmates. As n the case of the bvarate model the assumptons above we can say that E( ) = for all j =,..., k j j () We want to characterze the varance of our estmates. Whle t s possble to wrte down formulas here they are ecessvely complcated. Suffce to know that j σ Var( ) = f ( the ' s, ) for all j =,..., k (3) The dstrbuton of our estmates. It s gong to be normal for the same reason that the dstrbuton was normal n the bvarate case (assumpton 6). If the sample sze s small we need to make a normalty assumpton to conduct statstcal nference If the sample sze s large then we have a central lmt theorem that tells that the s are appromately normal. Note that we wll need an estmate of σ to conduct statstcal nference. We can estmateσ wth e SSE σ = = n ( k + ) n ( k + )
15 The FTest of a Set of Lnear Restrctons Eample : Human Captal Model. Are returns to potental eperence dfferent for men and women? There are two relevant models here Unrestrcted Model: ln( wage ) = + ed + e + esq + fem e + fem esq black + hsp + female + e Restrcted Model: ln( wage ) = + ed + e + esq black + 7 hsp + 8 female + e The restrctons mpled by the restrcted model are 4 = 5 =, therefore my hypothess test can be constructed as H : = = 4 5 H :, a 4 5
16 To evaluate the hypothess I must estmate both the restrcted and unrestrcted models Restrcted Model Estmates. reg lnwage ed e esq black hsp fem f year==987 Source SS df MS Number of obs = F( 6, 4993) = 447. Model Prob > F =. Resdual Rsquared = Adj Rsquared =.3488 Total Root MSE = lnwage Coef. Std. Err. t P> t [95% Conf. Interval] ed e esq black hsp fem _cons Unrestrcted Model Estmates. reg lnwage ed e esq fe fesq black hsp fem f year==987 Source SS df MS Number of obs = F( 8, 499) = Model Prob > F =. Resdual Rsquared = Adj Rsquared =.36 Total Root MSE = lnwage Coef. Std. Err. t P> t [95% Conf. Interval] ed e esq fe fesq black hsp fem _cons Comparson of Rsquared terms from the regressons. F R R * r = = 54 R.363 n ( k ) 4,99 +
17 Alternatvely, a seres of commands n STATA wll gve you the test statstc the pvalue assocated wth the test statstc.. reg lnwage ed e esq fe fesq black hsp fem f year==987. test fe== ( ) fe =. F(, 499) =.7 Prob > F =.. test fesq==, accum ( ) fe =. ( ) fesq =. F(, 499) = 53.7 Prob > F =.
18 Eample: Layoffs (From Chapter 6 of the tet). Do manageral and sales workers have dfferent epected unemployment duratons than productons workers? Age  age of the worker n years Educaton  hghest school grade completed Marred  a dummy varable equal to f worker s marred, otherwse Head  a dummy varable equal to f a worker s the head of household, otherwse Tenure The number of years on the last job Manager a dummy varable equal to f a worker was employed as a manager on hs last job, otherwse. Sales a dummy varable equal to f a worker was employed as a sales worker on hs last job, otherwse. Model: Weeks Unem = + Age + Educaton + Marred 3 + Head + Tenure + Manager + Sales + e Below s a verson of what STATA prnts out wth ts regresson tool.. reg weeks age educ marred head tenure manager sales Source SS df MS Number of obs = F( 7, 4) = 8.68 Model Prob > F =. Resdual Rsquared = Adj Rsquared =.533 Total Root MSE = weeks Coef. Std. Err. t P> t [95% Conf. Interval] age educ marred head tenure manager sales _cons
19 Weeks Unem = γ + γ Marred + γ Head + γ3 Tenure Restrcted Model: + γ Manager + γ Sales + e 4 5. reg weeks marred head tenure manager sales Source SS df MS Number of obs = F( 5, 44) = 4.59 Model Prob > F =.9 Resdual Rsquared = Adj Rsquared =.683 Total Root MSE = weeks Coef. Std. Err. t P> t [95% Conf. Interval] marred head tenure manager sales _cons * In actualty you would not need to estmate the restrcted model to test the hypothess that = =. In practce, the test statstc and pvalue assocated wth ths test obtaned wth the followng seres of commands n STATA. reg weeks age educ marred head tenure manager sales. test marred==. test educ==, accum
20 Eample: Unemployed Workers (From Chapter 6 of the tet) Age  age of the worker n years Educaton  hghest school grade completed Marred  a dummy varable equal to f worker s marred, otherwse Head  a dummy varable equal to f a worker s the head of household, otherwse Tenure The number of years on the last job Manager a dummy varable equal to f a worker was employed as a manager on hs last job, otherwse. Sales a dummy varable equal to f a worker was employed as a sales worker on hs last job, otherwse. Model: Weeks Unem = + Age + Educaton + Marred 3 + Head + Tenure + Manager + Sales + e
21 . reg weeks age educ marred head tenure manager sales Source SS df MS Number of obs = F( 7, 4) = 8.68 Model Prob > F =. Resdual Rsquared = Adj Rsquared =.533 Total Root MSE = weeks Coef. Std. Err. t P> t [95% Conf. Interval] age educ marred head tenure manager sales _cons How can we nterpret the coeffcent on the educaton varable? Lets test the hypothess that educaton has no effect on unemployment duraton of manufacturng workers. How would I state ths hypothess n terms of the populaton parameters? What should the frst step n testng ths hypothess be? What do I do net? What conclusons can I draw?
22 Testng a Set of Lnear Restrctons (The FTest) What I am gong to gve you s a more general verson of the FTest that s n the book. What I gve you s way more powerful that what s n the book so please pay attenton. Ths s open materal for the test. Consder the model y = kk + e Suppose that we want to test the hypothess that some subset of the coeffcents s zero. We can form an Fstatstc for ths test.
23 F * SSE SSE = r SSE F(, r n k) n k where r = the number of restrctons SSE * = the error sum of squares from a restrcted model SSE = the error sum of squares from the unrestrcted model Just for the sake of beng concrete lets suppose that y = e and want to test the null hypothess that = = aganst the alternatve that and. To test ths hypothess t turns out we wll need an F statstc. Ths Fstatstc can be computed as F * SSE SSE = r SSE n k Where SSE = e = ( y ) * SSE = e = ( y γ γ γ ) * 3 4
24 Where the γ s are least squares coeffcent estmates from the model y = γ + 3γ3 + γ4 + v The mportant thng about ths model s that t s a restrcted verson of the frst model. It s essentally the frst model wth the coeffcents on the frst two varables set to zero. If the null hypothess that = = s correct, I would epect that the sum of squared resduals from the restrcted model would not be much larger than the sum of squares from the unrestrcted model. It wll be larger because degrees of freedom due to fewer coeffcents, but t should not be much larger. There s an easy way to compute the test statstc F * SSE SSE = r SSE n k If I dvde the numerator and denomnator by SST I get F r SST SST ( R ) r r = = = SSE R R n k SST n k n k * SSE SSE R R * R * Bascally all I need to do to conduct the test s estmate both the restrcted and the unrestrcted models, get the Rsquareds from these regressons, form the Fstatstc and go to the tables. Eample: Unemployment Duratons Lets test the hypothess that the age and educaton jontly have not effect on weeks unemployed for lad off manufacturng workers. H H : = = a :, I need to compute estmates from the restrcted model
25 Weeks Unem = γ + γ Marred + γ Head + γ3 Tenure Model: + γ Manager + γ Sales + e 4 5. reg weeks marred head tenure manager sales Source SS df MS Number of obs = F( 5, 44) = 4.59 Model Prob > F =.9 Resdual Rsquared = Adj Rsquared =.683 Total Root MSE = weeks Coef. Std. Err. t P> t [95% Conf. Interval] marred head tenure manager sales _cons Alternatvely, a seres of commands n STATA wll gve you the test statstc the pvalue assocated wth the test statstc. reg weeks marred head tenure manager sales test marred== test educ==, accum
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