0. Introductory econometrics

Transcription

1 0. Introductory econometrics 0.1 Structure of economic data Cross-sectional data: Data wic are collected from units of te underlying population at a given time period (wic may vary occasionally) (te arrangement of te units in te dataset is irrelevant) Starting point is mostly te implicit assumption tat te data ave been collected by random sampling Examples: Individual or ouseold data (e.g. income), firm data (e.g. sales), city or country data (e.g. unemployment) Example: Observation number Country Population density GDP per capita Labor force (rural) GDP growt Birt rate Net migration 1 A B C : : : : : : : : 10 J K L

2 Time series data: Data wic are collected for one or more variables during several successive time periods Time is an important dimension (i.e. observations are often correlated over time) so tat te arrangement of te observations in te data set contains potentially important information Te frequency of data collection over time may vary strongly, e.g. daily, weekly, montly, quarterly, and annual data wit possible seasonal effects Examples: Macroeconomic data (e.g. income, consumption, investments, supply of money, price index), financial market data (e.g. stock prices) Example: Observation number Year Inflation US Unemployment rate US : : : :

3 Pooled cross-sectional data: Data wic ave cross-sectional as well as time series caracteristics since several cross-sectional data sets are collected independently over different time periods and linked to increase te sample size Altoug te arrangement of te observations in te data set is not essential, te corresponding period is recognized as an important variable Te data are mostly analyzed like conventional cross-sectional data Examples: Individual or ouseold data (e.g. income, expenditures) during several years Example: Observation number Year House price Wealt tax House size : : : : : : : : : :

4 Panel data: Data wic ave bot a time series and a cross-sectional dimension were (in contrast to pooled cross-sectional data) te same units (e.g. individuals, firms, countries) are observed over several time periods Te number of units is often muc iger tan te time dimension Te data is often first sorted by units and ten by periods Te data provide te opportunity to control for unobserved caracteristics of te units and to examine lagged variables Examples: Individual or ouseold panel data (e.g. SOEP), firm panel data (e.g. MIP), country panel data Example: Observation number Houseold Year Size Net income Smoking ouseold yes yes no no : : : : : : no no

5 0.2 Linear regression models (for cross-sectional data) Multiple linear regression model: y = β 0+ β1x 1+ β2x 2+ β3x 3+ + βk-1x k-1+ βkx k+ ε x 1, x 2, x 3,, x k-1, x k : Explanatory variables β 0 : Intercept β 1 : Tis parameter measures te effect of an increase of x 1 on y, olding all oter observed and unobserved factors fixed β 2 : Tis parameter measures te effect of an increase of x 2 on y, olding all oter observed and unobserved factors fixed : β k : Tis parameter measures te effect of an increase of x k on y, olding all oter observed and unobserved factors fixed ε: Error term Key assumption for te error term ε: E(ε x, x,, x ) = k Tis assumption states tat te error term ε is mean independent of te explanatory variables x 1, x 2,, x k. 5

6 For te furter analysis of linear regression models a sample of size n from te population is required. Multiple linear regression model wit k explanatory variables: {(x i1, x i2,, x ik, y i ), i = 1,, n} Te inclusion of te observations i = 1,..., n leads to te following linear regression model: y = β + β x + β x + + β x + ε i 0 1 i1 2 i2 k ik i For example, x ik is te value of explanatory variable k for observation i. Main task of regression analysis: Estimation of te unknown regression parameters β 0, β 1, β 2, Optimization problem in te ordinary metod of least squares (OLS metod) for te multiple linear regression model: n min (y - b - b x - b x - - b x ) b 0, b 1, b 2,..., bk i=1 i 0 1 i1 2 i2 k ik 2 6

7 Te first order conditions for te k+1 estimated regression parameters are ten: n i=1 n i=1 n i=1 (y - βˆ - βˆ x - βˆ x - - βˆ x ) = 0 i 0 1 i1 2 i2 k ik x (y - βˆ - βˆ x - βˆ x - - βˆ x ) = 0 i1 i 0 1 i1 2 i2 k ik x (y - βˆ - βˆ x - βˆ x - - βˆ x ) = 0 i2 i 0 1 i1 2 i2 k ik n i=1 x (y - βˆ - βˆ x - βˆ x - - βˆ x ) = 0 ik i 0 1 i1 2 i2 k ik OLS fitted values (predicted values) are te estimated values of te dependent variable: ŷ = β ˆ + βˆ x + βˆ x + + βˆ x for i = 1,, n i 0 1 i1 2 i2 k ik OLS regression equation: ŷ = β ˆ + β ˆ x + β ˆ x + + β ˆ x k k 7

8 Interpretation of te OLS estimated parameters in multiple linear regression models: ŷ = β ˆ + βˆ x + βˆ x + + βˆ x k k ŷ = βˆ x + βˆ x + + βˆ x k k If x 2, x 3, x 4,, x k are eld fixed, it follows: ŷ = βˆ x 1 1 In tis case, te estimated parameter of te explanatory variable x 1 tus indicates te cange of te estimated dependent variable if x 1 increases by one. If x 1, x 2, x 3,, x k-1 are eld fixed, it follows: Δy ˆ = βˆ Δx k k In tis case, te estimated parameter of te explanatory variable x k tus indicates te cange of te estimated dependent variable if x k increases by one. Terefore, te estimated parameters can be interpreted as estimated partial effects, i.e. te estimated effect of an explanatory variable implies tat it is controlled for all oter explanatory variables. Tis partial effect interpretation is te strong advantage of regression analyses (and of econometric analyses in general), i.e. a ceteris paribus interpretation is possible witout te necessity of conducting a corresponding controlled experiment. 8

9 Residuals (estimated error terms): Difference between te actual values of te dependent variable and te OLS fitted values ε ˆ = y - y ˆ = y - βˆ - βˆ x - βˆ x - - βˆ x for i = 1,, n i i i i 0 1 i1 2 i2 k ik Alternative formulation of linear regression models: y = y ˆ + ε ˆ = β ˆ + βˆ x + βˆ x + + βˆ x + ε ˆ for i = 1,, n i i i 0 1 i1 2 i2 k ik i Total sum of squares: n SST = (y - y) i=1 Explained sum of squares: Residual sum of squares (or sum of squared residuals): General rule: i 2 n n 2 2 ˆ ˆ i ˆ i i=1 i=1 SSE = (y - y) = (y - y) n n 2 2 ˆ ˆ i ˆi i=1 i=1 SSR = (ε - ε) = ε SST = SSE + SSR SSR SSE + = 1 SST SST 9

10 Coefficient of determination: Ratio between te explained variation and te total variation (of te dependent variable y i ) 2 SSE SSR R = = 1 - SST SST Te coefficient of determination also equals te squared correlation coefficient between te actual dependent variables and te OLS fitted values: n 2 n 2 (yi- y)(yˆ i- y) ˆ (yi- y)(yˆ i- y) 2 i=1 i=1 n n n n (y ˆ ˆ ˆ i- y) (yi- y) (yi- y) (yi- y) i=1 i=1 i=1 i=1 R = = Properties of te coefficient of determination: 0 R 2 1 R 2 never decreases, if an additional (and possibly irrelevant) explanatory variable is included (since SSR never rises in tis case) Terefore, R 2 is a poor measure to evaluate te quality of a linear regression model (also te adjusted coefficient of determination, wic takes te number of explanatory variables into account, is not an appropriate measure for evaluating te quality of a linear regression model) 10

11 Example: Determinants of (te logaritm of) wages (I) By using a linear regression model, te effect of te years of education (educ), te years of labor market experience (exper), and te years wit te current employer (tenure) on te logaritm of ourly wage (logwage) is examined: logwage = β 0+ β1educ + β2exper + β3tenure + ε Te following OLS regression equation was estimated on te basis of n = 526 workers: logwage ˆ = educ exper tenure Interpretation: Estimated positive effect of te years of education: If exper and tenure are eld fixed, an increase of te years of education by one year leads to an estimated increase of te logaritm of wage by Exper and tenure (as expected) ave also positive estimated effects, if te oter explanatory variables are eld fixed, respectively

12 Example: Determinants of (te logaritm of) wages (II) reg logwage educ exper tenure Source SS df MS Number of obs = F( 3, 522) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = logwage Coef. Std. Err. t P> t [95% Conf. Interval] educ exper tenure _cons

13 0.3 Expected value and variance of OLS estimators Assumptions for te analysis of te expected value of OLS estimators: Assumption A1: Linear in parameters Te relationsip between te dependent variable y and te explanatory variables x 1, x 2,, x k is linear in te parameters wit y = β 0 + β 1 x 1 + β 2 x β k x k + ε Assumption A2: Random sampling Te OLS estimation is based on a random sample wit n observations from te population wit {(x i1, x i2,, x ik, y i ), i = 1,, n} so tat it follows for a particular observation i: y i = β 0 + β 1 x i1 + β 2 x i2 + + β k x ik + ε i Assumption A3: No perfect collinearity In te sample (and terefore also in te population) none of te explanatory variables is constant and no exact linear relationsip between te explanatory variables exists Assumption A4: Zero conditional mean E(ε x 1, x 2,, x k ) = 0 Under tese four assumptions all OLS estimators are unbiased: E(β ˆ ) = β for = 0, 1,, k 13

14 Remark about assumption A4: If tis assumption olds, te explanatory variables are caracterized as exogenous. In contrast, if it is violated, endogenous variables or endogeneity are present. A4 is e.g. violated in te case of measurement errors in te explanatory variables or if te functional relationsip between te dependent and explanatory variables is misspecified One of te major violations of A4 is te omission of a relevant explanatory variable, wic is correlated wit oter explanatory variables Possible biases due to te omission of relevant explanatory variables ( omitted variable bias ) Te correct linear regression model is as follows (were assumptions A1 troug A4 old): y = β + β x + β x + + β x + β x + ε k-1 k-1 k k However, te following misspecified linear regression model tat omits x k is estimated (e.g. due to te lack of knowledge or te lack of data): y = β + β x + β x + + β x + ε k-1 k-1 Te correct and misspecified OLS regression equations are as follows: 14

15 ŷ = β ˆ + βˆ x + βˆ x + βˆ x + βˆ x k-1 k-1 k k y = β + β x + β x + β x k-1 k-1 In tis case, te following relationsip exists: β = β ˆ + βˆ δ k δ ( = 1,, k-1) is te OLS estimated slope parameter for x from a regression of x k on all oter explanatory variables (including a constant). It follows: E(β ) = β + βkδ Te OLS estimator of te slope parameter is tus usually biased, even wen te direction of te bias is ambiguous. Te estimator is only unbiased if β k or δ is zero. If δ is zero, x and x k are uncorrelated in te sample. In contrast: Te inclusion of irrelevant explanatory variables (i.e. one or more explanatory variables tat ave no partial effect on te dependent variable) as no impact on te unbiasedness of te OLS estimators and tus does not lead to biases However, te inclusion of irrelevant explanatory variables as an impact on te variance of te OLS estimators 15

16 Assumptions for te analysis of te variance of OLS estimators: Assumptions A1 troug A4 from te analysis of te expected value of OLS estimators Assumption A5: Homoskedasticity Te conditional variance of te error term ε is constant, i.e. Var(ε x 1, x 2,, x k ) = σ 2. If te assumption is violated, i.e. if te variance depends on te explanatory variables, tis leads to eteroskedasticity. Te assumptions A1 troug A5 are also known as te Gauss-Markov assumptions (in te case of regression analyses wit cross-sectional data) Under te assumptions A1 troug A5 te variance of te OLS estimated slope parameters in linear regression models is: 2 2 ˆ σ σ Var(β ) = = for = 1,, k n (1-R )SST (1-R ) (xi- x ) i=1 R 2 is te coefficient of determination from regressing x on all oter explanatory variables (including a constant). Wile te assumption of omoskedasticity is negligible for te unbiasedness of te estimated parameters, te above variance is only true wit te omoskedasticity assumption, but not in te case of eteroskedasticity 16

17 Estimation of te variance σ 2 of te error term ε: Te estimation of σ 2 is te basis for te estimation of te variance of te OLS estimated regression parameters Since σ 2 = E(ε 2 ), te following estimator would be obvious: n 1 2 SSR ˆε i = n i=1 n However, tis estimator is biased. In contrast, an unbiased estimator is te ratio between SSR and te difference between te sample size n and te number k+1 of regression parameters: 1 SSR ˆσ = ε = n-k-1 n 2 2 ˆi n-k-1 i=1 Tus, te corresponding (consistent, but not unbiased) standard error of te regression (SER), wic is an estimator of te standard deviation σ of te error term ε, is: σ ˆ = σ ˆ = 1 n 2 2 ε ˆi n-k-1 i=1 17

18 An unbiased estimator of te variance of te OLS estimated slope parameters in linear regression models is terefore: Var(β ˆ ˆ ) = 2 ˆσ (1-R )SST 2 for = 1,, k Standard deviation of te OLS estimated slope parameters: Var(β ˆ ) = σ (1-R )SST 2 for = 1,, k Tis standard deviation can ten be estimated as follows: Var(β ˆ ˆ ) = ˆσ (1-R )SST 2 for = 1,, k Te use of tese estimates (also called standard errors of te estimated parameters) is particularly based on te omoskedasticity assumption A5. In contrast, te variance of te OLS estimated slope parameters is estimated wit a bias in te case of eteroskedasticity (altoug eteroskedasticity as no influence on te unbiasedness of te estimated regression parameters). 18

19 Under te assumptions A1 troug A5 it follows: Te OLS estimators are te best linear unbiased estimators (BLUE) of te regression parameters in linear regression models Components of BLUE: Unbiased indicates tat te estimator is not biased Linear indicates tat te estimator is a linear function of te data and te dependent variable Best indicates tat te estimator as te smallest variance In accordance wit te Gauss-Markov teorem, OLS estimators ave te smallest variance in te class of all linear and unbiased estimators. However, te prerequisite for tis property is tat te assumptions A1 troug A5 old. 19

20 0.4 Testing of ypoteses about regression parameters Additional assumption A6: Normality Te error term ε is independent from te explanatory variables x 1, x 2,, x k and normally distributed wit an expected value of zero and a variance of σ 2 : ε ~ N(0; σ 2 ) Te assumptions A1 troug A6 are also called classical linear model assumptions. Terefore, te corresponding approac is also called classical linear regression model. Under te assumptions A1 troug A6 it follows for te dependent variable: 2 y x 1, x 2,, x k N(β 0+ β1x 1+ β2x 2+ + βkx k; σ ) It follows: Te OLS estimators are te best unbiased estimators (BUE) of te regression parameters in linear regression models. Te OLS estimators tus ave te smallest variance not only in te class of all linear unbiased estimators, but also in te larger class of all unbiased estimators. Indeed, if te error term ε is not normally distributed, te realization of statistical tests is no problem if te sample size n is large 20

21 If assumption A6 (normally distributed error term ε) olds, te OLS estimated slope parameters in linear regression models are also normally distributed, i.e. it follows ( = 1,, k): β ˆ N[β ; Var(β ˆ )] resp. β ˆ N β ; n 2 2 (1-R ) (xi- x ) i=1 It follows ( = 1,, k): βˆ - β Var(β ˆ ) βˆ - β N(0; 1) resp. N(0; 1) σ In addition, eac linear function of te OLS estimated regression parameters β 0, β 1,, β k is normally distributed. σ n 2 2 i i=1 (1-R ) (x - x ) 2 21

22 However, te variances and standard deviations of te OLS estimated slope parameters in linear regression models are usually unknown and tus ave to be estimated. Under te assumptions A1 troug A6 it follows: βˆ - β Var(β ˆ ˆ ) n-k-1 βˆ - β t resp. t ˆσ n 2 2 i i=1 (1-R ) (x - x ) k+1 is te number of unknown regression parameters. Te main null ypotesis tat is tested in empirical applications is: H 0: β = 0 for = 1,, k Te null ypotesis about te slope parameter β implies tat te explanatory variable x as no partial effect on te dependent variable y. Te test statistic in tis case is te following t statistic (t value), wic includes te estimated standard deviation (standard error) of te estimated parameters: t = t = t = βˆ ˆβ Var(β ˆ ˆ ) n-k-1 22

23 Te testing of H 0 : β = 0 at a given significance level is based on te property tat te t statistic is t distributed wit n-k-1 degrees of freedom under te null ypotesis. In empirical analyses a two-tailed test is usually examined. Te two-sided alternative ypotesis is: H 1: β 0 for = 1,, k Te null ypotesis is tus rejected if: t > t n-k-1;1-α/2 More general null ypotesis: H 0: β = a for = 1,, k Te null ypotesis is rejected if β is strongly different from a. Te appropriate test statistic is te following more general t statistic: t = ˆβ - a Var(β ˆ ˆ ) If H 0 : β = a is true, tis t statistic is again t distributed wit n-k-1 degrees of freedom. Te null ypotesis is rejected at te significance level α in favor of te alternative ypotesis H 1 : β a if t > t n-k-1;1-α/2. 23

24 Example: Effect of air pollution on ousing prices (I) By using a linear regression model, te effect of te logaritm of nitrogen oxides (in ppm) in te air (lognox), te logaritm of te weigted distances (in miles) from five employment centers (logdist), te average number of rooms in ouses (rooms), and te average ratio between teacers and pupils in scools (stratio) on te logaritm of te median ousing prices (logprice) is examined wit a sample of n = 506 communities. Te following OLS regression equation was estimated (R 2 = 0.584): logprice ˆ = lognox logdist rooms stratio (0.32) (0.117) (0.043) (0.019) (0.006) Due to te ig common t values, all explanatory variables ave an effect at common significance levels (e.g. 0.05, 0.01). Anoter interesting null ypotesis refers to te testing weter β 1 equals te value -1, i.e. H 0 : β 1 = -1. In tis case it follows t = ( )/0.117 = Terefore, te null ypotesis cannot be rejected at common significance levels (i.e. te estimated elasticity is not significantly different from te value -1)

25 Example: Effect of air pollution on ousing prices (II) reg logprice lognox logdist rooms stratio Source SS df MS Number of obs = F( 4, 501) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = logprice Coef. Std. Err. t P> t [95% Conf. Interval] lognox logdist rooms stratio _cons

26 Also ypoteses about linear combinations of regression parameters can be tested. Wit arbitrary values r 1, r 2,,r k and c te corresponding null ypotesis can be specified as follows: H 0: r1β 1+ r2β 2+ + rkβ k = c resp. H 0: r1β 1+ r2β 2+ + rkβk - c = 0 Te inclusion of te estimated variance of te linear combination of te slope parameters leads to te following t statistic, wic is t distributed wit n-k-1 degrees of freedom under te null ypotesis: t = r1β ˆ ˆ 1+ + rkβ k- c Var(r ˆ β ˆ + + r β ˆ ) 1 1 k k An often considered null ypotesis refers to te equality of two parameters, e.g.: H : β = β resp. H : β - β = Te corresponding t statistic is: t = βˆ ˆ 1-β2 Var(β ˆ ˆ -β ˆ ) 1 2 H 0 is tus rejected at te significance level α (in te case of a two-tailed test) if t > t n-k-1;1-α/2. 26

27 Finally, also multiple linear exclusion restrictions can be tested. Te starting point is te following (unrestricted) multiple linear regression model: y = β 0+ β1x 1+ β2x 2+ + βkx k+ ε In order to test weter a group of q explanatory variables as no effect on te dependent variable, te null ypotesis can be stated as follows: H 0: β k-q+1= 0, β k-q+2= 0,, β k= 0 resp. H 0: β k-q+1= β k-q+2= = β k = 0 Te restricted regression model under H 0 is ten: y = β 0+ β1x 1+ β2x 2+ + βk-qx k-q + ε For te corresponding F test te following F statistic (F value) is considered: SSR r- SSR ur q SSR r- SSR ur n-k-1 F = = SSR ur SSR ur q n-k-1 If H 0 : β k-q+1 = β k-q+2 = = β k = 0 is true, tis test statistic is F distributed wit q (i.e. te number of exclusion restrictions) and n-k-1 degrees of freedom, i.e.: F F q;n-k-1 Te null ypotesis is rejected at te significance level α in favor of te alternative ypotesis if F > F q;n-k-1;1-α. 27

28 Alternative formulation wit te coefficients of determination R 2 r and R 2 ur in te restricted and unrestricted linear regression models: 2 2 R ur- R r 2 2 q R ur- R r n-k-1 F = = R ur 1-R ur q n-k-1 Te most commonly considered F test in empirical analyses refers to te following null ypotesis: H 0: β 1= β 2= = β k= 0 Tis leads to te following restricted linear regression model: y = β + ε 0 Te coefficient of determination R 2 r is zero in suc restricted linear regression models so tat te (R-squared form of te) F statistic in tis case wit q = k restrictions is (wit R 2 as te ordinary coefficient of determination in a linear regression model wit k explanatory variables): 2 R F = k = R R 1-R k n-k-1 2 n-k-1 28

29 Example: Determinants of birt weigts (I) By using a linear regression model, te effect of te average number of cigarettes te moter smoked per day during pregnancy (cigs), te birt order of te cild (parity), te annual family income (faminc), te number of years of scooling for te moter (moteduc), and te number of years of scooling of te fater (fateduc) on te birt weigt (in ounces = grams) of cildren (bwgt) is examined: bwgt = β + β cigs + β parity + β faminc + β moteduc + β fateduc + ε On te basis of a significance level of 0.05, te null ypotesis tat te number of years of scooling of te parents as no effect on te birt weigt is considered, i.e. H 0 : β 4 = β 5 = 0: For n = 1191 birts te unrestricted and restricted regression models are estimated by OLS. It follows R 2 r = and R 2 ur = Since n-k-1 = = 1185 and q = 2 it follows for te F statistic: F = [( )/( )](1185/2) = 1.42 Te critical value from te F distribution wit 2 and 1185 degrees of freedom is F 2;1185;0.95 = Terefore, te null ypotesis cannot be rejected at te 5% significance level

30 Example: Determinants of birt weigts (II) reg bwgt cigs parity faminc moteduc fateduc Source SS df MS Number of obs = F( 5, 1185) = 9.55 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = bwgt Coef. Std. Err. t P> t [95% Conf. Interval] cigs parity faminc moteduc fateduc _cons

31 Example: Determinants of birt weigts (III) reg bwgt cigs parity faminc Source SS df MS Number of obs = F( 3, 1187) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = bwgt Coef. Std. Err. t P> t [95% Conf. Interval] cigs parity faminc _cons Testing command and results in STATA (only possible directly after te OLS estimation in te unrestricted regression model, differences are due to roundings): test moteduc=fateduc=0 ( 1) moteduc - fateduc = 0 ( 2) moteduc = 0 F( 2, 1185) = 1.44 Prob > F =

32 0.5 Asymptotic properties Definition of consistency: Let W n be te estimator of parameter θ based on a sample y 1, y 2,,y n. Ten W n is a consistent estimator of θ if P( W n θ > ξ) converges (for ξ > 0) to zero for n. In tis case W n converges stocastically to θ, i.e. plim(w n ) = θ. Consistency of OLS estimators: If te assumptions A1 troug A4 old, te OLS estimators β ( = 0,1,, k) in linear regression models are consistent estimators of β, i.e. plim(β ) = β For te consistency of OLS estimators te same assumptions as for te unbiasedness are terefore required, i.e. e.g. te assumption A5 (omoskedasticity) can be violated. In fact, for te consistency of OLS estimators already a weakening of A4 is sufficient in addition to te assumptions A1 troug A3, i.e. A4 : E(ε) = 0 and Cov(x, ε) = 0 ( = 1, 2,, k). Inconsistency of OLS estimators: Remember: If E(ε x 1, x 2,, x k ) 0, i.e. if A4 is violated, te OLS estimators in linear regression models are not unbiased In te same manner, all OLS estimators are inconsistent if ε is correlated wit any explanatory variable, i.e. if A4 is violated 32

33 Asymptotic distributions of OLS estimators: Te exact normal distribution of te OLS estimators in linear regression models (and terefore te exact t and F distributions of te t and F statistics) is based on assumption A6, i.e. ε ~ N(0; σ 2 ). However, functions of te OLS estimators can also be asymptotically normally distributed if A6 is violated. If te assumptions A1 troug A5 old, it follows for te OLS estimated slope parameters in linear regression models (even witout assumption A6): ˆβ - β Var(β ˆ ˆ ) a N(0; 1) Tis property does not contradict te previous property according to wic tis function is exactly t distributed wit n-k-1 degrees of freedom if te assumptions A1 troug A6 old since also te following notation is feasible (since te t distribution converges to te standard normal distribution if te number of degrees of freedom increases): ˆβ - β Var(β ˆ ˆ ) a t n-k-1 33

34 It follows: Even for te case tat te error term ε is not normally distributed, te previously considered t and F tests can be conducted and te confidence intervals can be constructed. Te prerequisite for tis is tat te sample size n is sufficiently large. For small n (or a small number of degrees of freedom n-k-1), e.g. te approximation of te t statistic towards te t distribution is insufficient. Asymptotic efficiency: Under te Gauss-Markov assumptions (and tus wit te assumptions A1 troug A5), OLS estimators β ( = 0, 1,, k) are asymptotically efficient in a class of consistent estimators β of te regression parameters in linear regression models, i.e. it follows for te asymptotic variance Avar: Avar[ n(βˆ -β )] Avar[ n(β -β )] 34

35 0.6 Te structure of dependent and explanatory variables Logaritmized und squared variables: Linear regression models can comprise non-linear relationsips by including (naturally) logaritmized und squared variables Overview of te inclusion of logaritmized variables: Linear regression model Dependent variable Explanatory variable Interpretation of te estimated slope parameter Level-level y x y = β x Level-log y logx y (β /100)% x Log-level logy x % y (100β ) x Log-log logy logx % y = β % x 35

36 Example: Effect of air pollution on ousing prices By using a linear regression model, te effect of te logaritm of nitrogen oxides (in ppm) in te air (lognox) and te average number of rooms in ouses (rooms) on te logaritm of te median ousing prices (logprice) is examined wit a sample of n = 506 communities. Te OLS estimation wit STATA leads to te following results (R 2 = 0.514): reg logprice lognox rooms logprice Coef. Std. Err. t P> t [95% Conf. Interval] lognox rooms _cons It follows: An increase of nitrogen oxides by 1% (i.e. % nox = 1) leads to an estimated reduction of te median ousing prices by 0.718% (if te variable rooms is eld fixed) An increase of te average number of rooms by one (i.e. rooms = 1) leads to an approximately estimated increase of te median of te real estate prices by = 30.6% (if nox is eld fixed)

37 Squared explanatory variables: Tese variables allow for increasing or decreasing (partial) marginal effects in linear regression models Remember: If y is regressed on x, te estimated parameter β indicates te cange of te OLS fitted value y if x increases by one unit (and all oter explanatory variables are eld fixed). As a consequence, te (partial) marginal effect is constant and does not depend on x. Additional inclusion of a squared explanatory variable x 1 2 (besides te k-1 explanatory variables x 1, x 2,, x k-1 ): y = β + β x + β x + β x + + β x + β x + ε k-1 k-2 k k-1 In tis case β 1 does not indicate alone te cange of y wit respect to x 1. Te OLS regression equation is: ŷ = β ˆ + βˆ x + βˆ x + βˆ x + + βˆ x + βˆ x k-1 k-2 k k-1 If x 2,, x k-1 are eld constant, it follows te approximation: ˆ ˆ ŷ ŷ (β +2β x ) x resp. β ˆ + 2βˆ x x1 Terefore, te estimated (partial) marginal effect of x 1 on y also depends on β 2 and te values of x 1. 37

38 Interaction terms: Tese terms allow tat te (partial) effect (or elasticity or semi elasticity) of an explanatory variable in linear regression models depends on different values of anoter explanatory variable Additional inclusion of an interaction term for x 1 and x 2 (besides te k-1 explanatory variables x 1, x 2,, x k-1 ): y = β + β x + β x + β x x + β x + β x + β x + ε k-1 k-2 k k-1 Again, in tis case β 1 does not indicate alone te cange of y wit respect to x 1. Te OLS regression equation is : ŷ = β ˆ + βˆ x + βˆ x + βˆ x x + βˆ x + + βˆ x + βˆ x k-1 k-2 k k-1 If x 2,, x k-1 are eld constant, it follows: ˆ ˆ ŷ ŷ = (β +β x ) x resp. = β ˆ + βˆ x x1 Terefore, te estimated (partial) marginal effect of x 1 on y also depends on β 3 and x 2. In tis case, interesting values of x 2 are generally examined (e.g. te mean of x 2 in te sample). β 1 alone only indicates te estimated (partial) effect of x 1 if x 2 is zero. 38

39 Qualitative explanatory variables: So far, te focus is implicitly on quantitative continuous dependent and explanatory variables in linear regression models (wit an unrestricted range) suc as wages, prices, or sales. However, in empirical analyses qualitative explanatory variables often play an important role suc as gender, race, sectoral effects, regional effects. Qualitative variables: Qualitative information on explanatory variables can be captured by corresponding binary (or dummy) variables tat ave exactly two possible categories and tus take two values, namely one and zero Te OLS estimation and te testing of ypoteses about regression parameters in linear regression models wit qualitative explanatory variables is fully equivalent to te exclusive inclusion of quantitative explanatory variables Single binary explanatory variables: Inclusion of qualitative variables wit two categories On te basis of a multiple linear regression model wit only quantitative explanatory variables, an additional binary explanatory variable x 0 is included (besides now k-1 quantitative explanatory variables x 1, x 2,, x k-1 ): 39

40 y = β 0+ β1x 0+ β2x 1+ β3x 2+ + βkx k-1+ ε Wit E(ε x 0,x 1,x 2,, x k-1 ) = 0 it follows: E(y x 0, x 1, x 2,, x k-1) = β 0+ β1x 0+ β2x 1+ β3x 2+ + βkxk-1 It follows: β = E(y x = 1, x, x,, x ) - E(y x = 0, x, x,, x ) k k-1 β 1 tus is te difference in te expected value of y between x 0 = 1 und x 0 = 0, given te same values of x 1, x 2,, x k-1 and ε. β 0 is terefore te constant for x 0 = 0. For x 0 = 1 te constant is β 0 + β 1, so tat β 1 is te difference of te constant for x 0 = 1 und x 0 = 0. Note: For one factor (e.g. gender) it is not possible to jointly include two dummy variables (e.g. one variable tat takes te value one for women and one variable tat takes te value one for men) in linear regression models since tis would lead to perfect collinearity (simple version of dummy variable trap ) 40

41 Example: Determinants of (te logaritm of) wages By using a linear regression model, te effects of gender (female), te years of education (educ), te years of labor market experience (exper), te squared years of labor market experience (expersq), te years wit te current employer (tenure) and te squared years wit te current employer (tenuresq) on te logaritm of ourly wage (logwage) is examined. On te basis of n = 526 workers, te OLS estimation wit STATA leads to te following results (R 2 = 0.441): reg logwage female educ exper expersq tenure tenuresq logwage Coef. Std. Err. t P> t [95% Conf. Interval] female educ exper expersq tenure tenuresq _cons Te results imply tat te estimated ourly wage for women is on average approximately = 29.7% smaller (for equal education, labor market experience, and years wit te current employer)

42 Binary explanatory variables for multiple categories: Inclusion of qualitative variables wit more tan two categories Te basis is again a multiple linear regression models wit only quantitative variables. Now an additional qualitative (nominal or ordinal) explanatory variable (e.g. sector, region, education) wit q > 2 different categories is considered. In tis case (maximal) q-1 dummy variables x 01, x 02,, x 0,q-1 can be included (besides now k-q+1 quantitative explanatory variables x 1, x 2,, x k-q+1 ): y = β + β x + β x + + β x + β x + β x + + β x + ε Category q of te qualitative variable (i.e. te dummy variable x 0q ) is considered as te base category. As a consequence, te estimated slope parameters β 1, β 2,, β q-1 indicate for te corresponding group of te qualitative variable (i.e. for x 01, x 02,, x 0,q-1 ) te estimated average difference in te dependent variable y compared wit te base category, i.e. compared wit x 0q. Note: q-1 0,q-1 q 1 q+1 2 k k-q+1 It is not possible to jointly include all q dummy variables x 01, x 02,, x 0q since tis would lead to perfect collinearity (general version of dummy variable trap ). Many econometric packages suc as STATA automatically correct for tis mistake. 42

43 Interaction terms wit binary explanatory variables: Interaction terms can also comprise dummy variables (and tus need not only refer to two quantitative explanatory variables) Additional inclusion of an interaction term for two binary explanatory variables x 01 und x 02 (besides now k-3 quantitative explanatory variables x 1, x 2,, x k-3 ): y = β 0+ β1x 01+ β2x 02+ β3x01x 02+ β4x 1+ β5x 2+ + βkx k-3+ ε Interpretation: Te inclusion of tese interaction terms (besides te separate inclusion of te corresponding dummy variables) is an alternative to te inclusion of tree binary explanatory variables if four categories are examined β 1 (resp. β 2 ) indicates for x 02 = 0 (resp. x 01 = 0) te estimated average difference in te dependent variable y between x 01 = 1 and x 01 = 0 (resp. between x 02 = 1 and x 02 = 0) For x 01 = 1 and x 02 = 0 (resp. for x 01 = 0 and x 02 = 1) te estimated constant is β 0 + β 1 (resp. β 0 + β 2 ) For x 01 = 1 and x 02 = 1 te estimated constant is β 0 + β 1 + β 2 + β 3 43

44 Inclusion of an interaction term for one binary explanatory variables x 0 and one quantitative explanatory variable x 1 (besides now k-2 quantitative explanatory variables x 1, x 2,, x k-2 ): y = β 0+ β1x 0+ β2x 1+ β3x0x 1+ β4x 2+ + βkx k-2+ ε Interpretation: Tese interaction terms allow te analysis of possible differences in te (partial) effect (or elasticity or semi elasticity) of te quantitative explanatory variable x 1 in linear regression models for te two categories of te binary explanatory variable x 0. If β 3 = 0, ten tere is no difference. If x 0 = 0, it follows for te OLS regression equation: ŷ = β ˆ + β ˆ x + β ˆ x + + β ˆ x k k-2 Te estimated constant in tis case is β 0 and te estimated (partial) effect of x 1 is β 2. If x 0 = 1, it follows for te OLS regression equation: ŷ = β ˆ + β ˆ + β ˆ x + β ˆ x + β ˆ x + + β ˆ x k k-2 Te estimated constant in tis case is β 0 + β 1 and te estimated (partial) effect of x 1 is β 2 + β 3. 44

45 0.7 Heteroskedasticity Previously, for te variance of te OLS estimators assumption A5 (omoskedasticity) was discussed in detail: If Var(ε x 1, x 2,, x k ) σ 2, tis leads to eteroskedasticity Unlike e.g. omitting relevant explanatory variables, eteroskedasticity as no impact on te unbiasedness or consistency of OLS estimators. However, eteroskedasticity as an impact on te (estimated) variance of te OLS estimated slope parameters in linear regression models. As discussed above, it follows in te case of omoskedasticity, i.e. under te assumptions A1 troug A5, for te variance of te OLS estimated slope parameters (wit R 2 as te coefficient of determination of a regression of x on all oter explanatory variables): 2 2 ˆ σ σ Var(β ) = = for = 1,, k n (1-R )SST (1-R ) (xi- x ) i=1 Tus, in te case of omoskedasticity, it follows te following estimated standard deviation wit a consistent estimator of te standard deviation σ: Var(β ˆ ˆ ) = ˆσ (1-R )SST 2 for = 1,, k 45

46 Since tis variance is only true in te case of omoskedasticity, but not under eteroskedasticity, tis estimated standard deviation is a biased and inconsistent estimator of te standard deviation of te OLS estimators In te case of eteroskedasticity te estimated standard deviations are tus no longer valid for constructing confidence intervals as well as t and F statistics. Te corresponding t statistics are terefore no longer t distributed (even for a large sample size n) and te corresponding F statistics are no longer F distributed in te case of eteroskedasticity. Finally, te desirable BLUE property (or efficiency) of OLS estimators and te property of asymptotic efficiency are not valid in te case of eteroskedasticity. However, by knowing te form of eteroskedasticity, it is possible to construct more efficient estimators compared to te OLS estimators. A standard approac to test omoskedasticity is te Breusc-Pagan test (an alternative is te Wite test). Te null ypotesis is: H 0: Var(ε x 1, x 2,,x k) = σ resp. H 0: E(ε x 1, x 2,,x k) = E(ε ) = σ If H 0 is not true, ε 2 is a function of one or more explanatory variables. If a linear function of all explanatory variables is considered, it follows in tis case wit an error term v wit a (conditional) expected value of zero: 2 ε = δ 0+ δ1x 1+ δ2x 2+ k k + δ x + v 46

47 Te null ypotesis of omoskedasticity is ten: H : δ = δ = = δ = k Since te ε i are unknown, tey are replaced by te corresponding estimators, i.e. te residuals ε i, so tat te squared residuals are regressed on te explanatory variables: 2 ˆε = δ 0+ δ1x 1+ δ2x 2+ k k A ig value of te coefficient of determination R 2 ε 2 in tis auxiliary regression suggests te validity of te alternative ypotesis, i.e. eteroskedasticity. One version of te Breusc Pagan test statistic is: BP = nr ˆε δ x + v Under te null ypotesis (i.e. omoskedasticity) BP is asymptotically χ 2 distributed wit k degrees of freedom, i.e.: a 2 BP χ k Tus, te null ypotesis of omoskedasticity is (for a large sample size n) rejected in favor of te alternative ypotesis of eteroskedasticity at te significance level α if: 2 BP > χ k;1-α 47

48 Example: Determinants of ouse prices (I) By using a linear regression model, te effect of te lot size in square feet (lotsize), te living space size in square feet (sqrft), and te number of bedrooms (bdrms) on ouse prices in 1000 dollar (price) is examined. Te following OLS regression equation was estimated: ˆ price = lotsize sqrft bdrms (29.48) ( ) (0.013) (9.01) 2 n = 88; R = On te basis of a Breusc Pagan test, te null ypotesis of omoskedasticity is tested at te 1% significance level: First, te residuals ε i are calculated. Te auxiliary regression of ε 2 on lotsize, sqrft, and bdrms leads to a coefficient of determination in te amount of R 2 ε 2 = Te corresponding Breusc Pagan test statistic amounts to te value of BP = = Wit k = 3 te critical value is χ 2 3;0.99 = Tus, te null ypotesis is rejected at te 1% significance level (te corresponding p-value is p = )

49 Example: Determinants of ouse prices (II) reg price lotsize sqrft bdrms Source SS df MS Number of obs = F( 3, 84) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = price Coef. Std. Err. t P> t [95% Conf. Interval] lotsize sqrft bdrms _cons Testing command and results in STATA (only possible directly after te OLS estimation): estat ettest, rs iid Breusc-Pagan / Cook-Weisberg test for eteroskedasticity Ho: Constant variance Variables: lotsize sqrft bdrms ci2(3) = Prob > ci2 =

50 If te null ypotesis is rejected at a low significance level and tus eteroskedasticity is verified, tis sould be considered: One possibility is te use of alternative estimation metods instead of te OLS metod suc as te weigted least squares (WLS) metod. However, for tis estimation metod, it is necessary to know te precise form of eteroskedasticity. In te case of eteroskedasticity te general question arises weter an alternative estimation metod instead of te OLS estimation sould really be applied: Since te OLS estimators are also unbiased and consistent in te case of eteroskedasticity (under te assumptions A1 troug A4), te use of OLS can still be useful. However, for te construction of confidence intervals and te application of t or F tests in te case of eteroskedasticity, te estimated standard deviations of te OLS estimators sould at least be corrected Te starting point of tese corrections are te actual (unknown) variances of te OLS estimators. Te unknown variances σ i 2 are replaced by te corresponding squared residuals ε i 2 (wic stem from te original OLS estimation). In multiple linear regression models te estimated variance of te OLS estimated slope parameters generally is: 50

51 Var(β ˆ ˆ ) = n 2 2 rˆˆ iεi i=1 2 SSR r i denotes te residual of observation i, wic stems from te regression of x on all oter explanatory variables, and SSR denotes te sum of squared residuals from tis regression. Te estimated standard deviation of te OLS estimated slope parameters according to Wite (1980) is ten: Var(β ˆ ˆ ) = n i=1 rˆˆ ε SSR 2 2 i i On tis basis several furter asymptotically equivalent estimators of standard deviations ave been developed. By using tese estimated standard deviations, eteroskedasticity robust confidence intervals and especially eteroskedasticity robust t statistics can be constructed. 51

52 Example: Determinants of (te logaritm of) wages (I) By using a linear regression model, te effect of te years of education (educ), te years of labor market experience (exper), te squared years of labor market experience (expersq), te years wit te current employer (tenure), te squared years wit te current employer (tenuresq), as well as tree combined variables for marital and gender status for married men (marrmale), married women (marrfem), and non-married women (singfem) on te logaritm of ourly wage (logwage) is examined. On te basis of n = 526 workers, te following OLS regression equation was estimated wic also reports eteroskedasticity robust estimated standard deviations of te estimated parameters (in brackets) in addition to conventionally estimated standard deviations (wit R 2 = 0.461): ˆ logwage = marrmale marrfem singfem educ (0.100) (0.055) (0.058) (0.056) (0.0067) [0.109] [0.057] [0.059] [0.057] [0.0074] exper expersq tenure tenuresq (0.0055) ( ) (0.0068) ( ) [0.0051] [ ] [0.0069] [ ]