Applied Econometrics (QEM)

Transcription

1 Applied Econometrics (QEM) based on Prinicples of Econometrics Jakub Mućk Department of Quantitative Economics Jakub Mućk Applied Econometrics (QEM) Meeting #3 1 / 42

2 Outline t-test P-value Linear combination of Parameters Normality of the error term 4 5 R 2 Adjusted R 2 Information Criteria Jakub Mućk Applied Econometrics (QEM) Meeting #3 2 / 42

3 Simple and Multiple Regression Model In simple linear model we consider only one explanatory variable x, i.e., where ε is the error term. y = β 0 + β 1 x + ε (1) In multiple regression model we can consider a set of explanatory variables x 1,..., x K, i.e. y = β 0 + β 1 x 1 + β 2 x β K x K + ε (2) A single parameter β i, measures the effect of a change in the variable x i upon the expected value of y, all other variables held constant: E (y) β i = E (y) x i =. (3) other xs held constant x i Jakub Mućk Applied Econometrics (QEM) Meeting #3 3 / 42

4 Simple and Multiple Regression Model In simple linear model we consider only one explanatory variable x, i.e., where ε is the error term. y = β 0 + β 1 x + ε (1) In multiple regression model we can consider a set of explanatory variables x 1,..., x K, i.e. y = β 0 + β 1 x 1 + β 2 x β K x K + ε (2) A single parameter β i, measures the effect of a change in the variable x i upon the expected value of y, all other variables held constant: E (y) β i = E (y) x i =. (3) other xs held constant x i Jakub Mućk Applied Econometrics (QEM) Meeting #3 3 / 42

5 Simple and Multiple Regression Model In simple linear model we consider only one explanatory variable x, i.e., where ε is the error term. y = β 0 + β 1 x + ε (1) In multiple regression model we can consider a set of explanatory variables x 1,..., x K, i.e. y = β 0 + β 1 x 1 + β 2 x β K x K + ε (2) A single parameter β i, measures the effect of a change in the variable x i upon the expected value of y, all other variables held constant: E (y) β i = E (y) x i =. (3) other xs held constant x i Jakub Mućk Applied Econometrics (QEM) Meeting #3 3 / 42

6 Assumptions of Multiple Regression Model I [Assumption #1] True DGP (data generating process): y = β 0 + β 1 x 1 + β 2 x β K x K + ε. (4) [Assumption #2] The expected value of the error term equals zero: E(ε) = 0 E (y) = β 0 + β 1 x 1 + β 2 x β K x K. (5) [Assumption #3] Variance of the error term (and dependent variable) var (ε) = var (y) = σ 2. (6) [Assumption #4] No dependence between errors: cov (ε i, ε j ) = cov (y i, y j ) = 0. (7) [Assumption #5] The values of each explanatory variable x i t are not random (exogeneity). Jakub Mućk Applied Econometrics (QEM) Meeting #3 4 / 42

7 Assumptions of Multiple Regression Model II It is assumed that the values of the explanatory variables are known to us prior to our observing the values of the dependent variable. [Assumption #6] The values of each explanatory variable x i t are not exact linear function of the other explanatory variables (no collinearity).. This assumption is equivalent to assuming that no variable is redundant. If this assumption is not satisfied (the exact collinearity case) then the least squares estimation cannot be performed. [Assumption #7] Normally distributed error term: ε i N ( 0, σ 2) y i N ( (β 0 + β 1 x 1 + β 2 x β K x K ), σ 2) (8) Jakub Mućk Applied Econometrics (QEM) Meeting #3 5 / 42

8 The least square estimator More generally, the least square estimator is obtained by minimizing the sum of squares (SSE), which is a function of the unknown parameters, given the data: SSE (β 0, β 1,..., β K ) = = N [y i E (y i )] 2 i=1 N [y i β 0 β 1 x 1... β K x K ] 2. i=1 Jakub Mućk Applied Econometrics (QEM) Meeting #3 6 / 42

9 Gauss-Markov Theorem Under the assumptions A#1-A#6 of the multiple linear regression LS LS LS model, the least squares estimators ˆβ 0, ˆβ 1,..., ˆβ K have the smallest variance of all linear and unbiased estimators of β 0, β 1,..., β K. ˆβ 0 LS LS, ˆβ ˆβ LS K 1,..., of β 0, β 1,..., β K. are the Best Linear Unbiased Estimators (BLUE) Jakub Mućk Applied Econometrics (QEM) Meeting #3 7 / 42

10 Outline t-test P-value Linear combination of Parameters Normality of the error term 4 5 R 2 Adjusted R 2 Information Criteria Jakub Mućk Applied Econometrics (QEM) Meeting #3 8 / 42

11 Point vs Point estimate is a single value of the estimator (mean). provides a range of values in which the true parameter is likely to fall allows to account for the precision with which the unknown parameter is estimated. The precision is typically measured with variance. Jakub Mućk Applied Econometrics (QEM) Meeting #3 9 / 42

12 I Under the assumption of normality of the error term the least squares estimator of ˆβ LS is: ˆβ LS N (β, Σ) (9) where Σ is the variance-covariance of the least squares estimator. For illustrative purpose we focus on slope parameter in the simple regression model ( ˆβ 1 ): LS ( ) ˆβ 1 LS σ 2 N β 1, N i (x i x) 2 (10) A standardized normal random variable can be obtained from subtracting its mean and dividing by its standard deviation ˆβ LS 1 by Z = ˆβ 1 LS β 1 σ N (0, 1). (11) 2 N i (x i x) 2 Jakub Mućk Applied Econometrics (QEM) Meeting #3 10 / 42

13 II Based on the features of standard normal distribution: we can substitute Z P 1.96 P ( 1.96 Z 1.96) =.95, (12) ˆβ LS 1 β 1 σ 2 N i (x i x) =.95, (13) and after manipulations: N P ˆβLS σ 2 (x i x) 2 β 1 i σ 2 ˆβ LS N (x i x) 2 =.95 i (14) Jakub Mućk Applied Econometrics (QEM) Meeting #3 11 / 42

14 III The two end-points estimator. LS ˆβ 1 ± ˆ1.96 σ 2 N i (x i x) 2 provide an interval In repeated sampling 95% of the intervals constructed this way will contain the true value of the parameter β 1. This easy derivation of an interval estimator is based on the assumption about normality of the error term and that we know the variance of the error term σ 2. Jakub Mućk Applied Econometrics (QEM) Meeting #3 12 / 42

15 Obtaining interval estimates I Replacing σ 2 by its estimates ˆσ 2 produces a random variable t: t = ˆβ LS 1 β 1 ˆσ 2 N i (x i x) 2 = ˆβ LS 1 β 1 ) = ˆvar ( ˆβLS 1 ˆβ LS 1 β 1 se ( ) β1 LS. (15) LS In multiple regression model, the t ratio, i.e. t = ( ˆβ 1 β 1 )/se(β1 LS ) has a t-distribution with N (K + 1) degrees of freedoms: where K is the number of explanatory variables. t t N (K+1), (16) The t-distribution is a bell shaped curve centered at zero. It looks like the standard normal distribution but it is more spread out, with a larger variance and thicker tails. The shape of the t-distribution is determined by a single parameter called the degrees of freedom. Jakub Mućk Applied Econometrics (QEM) Meeting #3 13 / 42

16 t- distribution t m f(t m ) t Jakub Mućk Applied Econometrics (QEM) Meeting #3 14 / 42

17 t- distribution t m f(t m ) α 2 α 2 t Jakub Mućk Applied Econometrics (QEM) Meeting #3 14 / 42

18 t- distribution t m f(t m ) α 2 α 2 1 α t Jakub Mućk Applied Econometrics (QEM) Meeting #3 14 / 42

19 Obtaining interval estimates I Critical value from a t distribution t c can be found as follows: P (t t c ) = P (t t c ) = α 2, (17) where α is arbitrary probability (significance level). The confidence intervals: P ( t c t t c ) = 1 α, (18) and after manipulations (with definition of t random variable): ( P ˆβLS 1 t c se ( β1 LS ) LS β1 ˆβ 1 + t c se ( β1 LS ) ) (19) Jakub Mućk Applied Econometrics (QEM) Meeting #3 15 / 42

20 Outline t-test P-value Linear combination of Parameters Normality of the error term t-test P-value Linear combination of Parameters Normality of the error term 4 5 R 2 Adjusted R 2 Information Criteria Jakub Mućk Applied Econometrics (QEM) Meeting #3 16 / 42

21 t-test P-value Linear combination of Parameters Normality of the error term Hypothesis testing is a comparison of a conjecture we have about a population to the information contained in a sample of data. The hypotheses are formed about economic behavior. In statistical inference, the hypotheses are then represented as statements about model parameters. General procedures: 1 A null hypothesis H 0, 2 An alternative hypothesis H 1, 3 A test statistic 4 A rejection region 5 A conclusion Jakub Mućk Applied Econometrics (QEM) Meeting #3 17 / 42

22 t-test P-value Linear combination of Parameters Normality of the error term Hypothesis testing is a comparison of a conjecture we have about a population to the information contained in a sample of data. The hypotheses are formed about economic behavior. In statistical inference, the hypotheses are then represented as statements about model parameters. General procedures: 1 A null hypothesis H 0, 2 An alternative hypothesis H 1, 3 A test statistic 4 A rejection region 5 A conclusion Jakub Mućk Applied Econometrics (QEM) Meeting #3 17 / 42

23 Null and alternative hypothesis t-test P-value Linear combination of Parameters Normality of the error term A null hypothesis is our belief we will maintain until it is not supported by the sample evidence, in which case we (can) reject the null hypothesis For instance, H 0 : β = c where c is an important value in the context of area of interest. An alternative hypothesis can be accepted if the H 0 can be rejected. Possible alternative hypotheses: H 1 : β c, H 1 : β c, H 1 : β c Jakub Mućk Applied Econometrics (QEM) Meeting #3 18 / 42

24 Null and alternative hypothesis t-test P-value Linear combination of Parameters Normality of the error term A null hypothesis is our belief we will maintain until it is not supported by the sample evidence, in which case we (can) reject the null hypothesis For instance, H 0 : β = c where c is an important value in the context of area of interest. An alternative hypothesis can be accepted if the H 0 can be rejected. Possible alternative hypotheses: H 1 : β c, H 1 : β c, H 1 : β c Jakub Mućk Applied Econometrics (QEM) Meeting #3 18 / 42

25 t-test P-value Linear combination of Parameters Normality of the error term The test statistics and rejection region Based on the value of a test statistic we decide either to reject the null hypothesis or not to reject it. The rejection region consists of values that are unlikely and that have low probability of occurring when the null hypothesis is true. The rejection region depends on: Distribution of test statistics when the null is true. Alternative hypothesis. Level of significance. Jakub Mućk Applied Econometrics (QEM) Meeting #3 19 / 42

26 t-test P-value Linear combination of Parameters Normality of the error term The test statistics and rejection region Based on the value of a test statistic we decide either to reject the null hypothesis or not to reject it. The rejection region consists of values that are unlikely and that have low probability of occurring when the null hypothesis is true. The rejection region depends on: Distribution of test statistics when the null is true. Alternative hypothesis. Level of significance. Jakub Mućk Applied Econometrics (QEM) Meeting #3 19 / 42

27 t-test P-value Linear combination of Parameters Normality of the error term The test statistics and rejection region Based on the value of a test statistic we decide either to reject the null hypothesis or not to reject it. The rejection region consists of values that are unlikely and that have low probability of occurring when the null hypothesis is true. The rejection region depends on: Distribution of test statistics when the null is true. Alternative hypothesis. Level of significance. Jakub Mućk Applied Econometrics (QEM) Meeting #3 19 / 42

28 t-test P-value Linear combination of Parameters Normality of the error term Type I & Type II error and significance level Type I error is a situation, in which we reject the null hypothesis when it is true. Type II error is a situation, in which we do not reject the null hypothesis when it is false. Significance level α: α is usually chosen to be 0.01, 0.05 or 0.10 P (Type I error) = α. (20) Jakub Mućk Applied Econometrics (QEM) Meeting #3 20 / 42

29 t-test P-value Linear combination of Parameters Normality of the error term Type I & Type II error and significance level Type I error is a situation, in which we reject the null hypothesis when it is true. Type II error is a situation, in which we do not reject the null hypothesis when it is false. Significance level α: α is usually chosen to be 0.01, 0.05 or 0.10 P (Type I error) = α. (20) Jakub Mućk Applied Econometrics (QEM) Meeting #3 20 / 42

30 t-test P-value Linear combination of Parameters Normality of the error term t-test and possible alternative hypotheses Based on the t statistics: t = ˆβ LS i β 1 se ( ) βi LS t N (K+1). (21) we can consider the following alternative hypotheses: 1 H 1 : β i c, 2 H 1 : β i c, 3 H 1 : β i c. Test of significance bases on the ratio with the null H 1 : β i = 0 and the alternative H 0 : β i 0. Jakub Mućk Applied Econometrics (QEM) Meeting #3 21 / 42

31 t-test P-value Linear combination of Parameters Normality of the error term One-tail test with alternative greater than The null and alternative: f(t m ) H 0 : β i = c H 1 : β i c t c = t 1 α,n (K+1) t α The null hypothesis can be rejected if t t (1 α,n (K+1)) Jakub Mućk Applied Econometrics (QEM) Meeting #3 22 / 42

32 t-test P-value Linear combination of Parameters Normality of the error term One-tail test with alternative less than The null and alternative: f(t m ) H 0 : β i = c H 1 : β i c α t c = t α,n (K+1) t The null hypothesis can be rejected if t t (α,n (K+1)) Jakub Mućk Applied Econometrics (QEM) Meeting #3 23 / 42

33 t-test P-value Linear combination of Parameters Normality of the error term Two-tail test with alternative not equal to The null and alternative: t m H 0 : β i = c f(t m ) α 2 t c = t α 2,N (K+1) t c = t 1 α,n (K+1) 2 α 2 H 1 : β i c The null hypothesis can be rejected if t t (α/2,n (K+1)) or t t (1 α/2,n (K+1)) t Jakub Mućk Applied Econometrics (QEM) Meeting #3 24 / 42

34 P-value t-test P-value Linear combination of Parameters Normality of the error term Standard practice is to use the probability value (p-value) This is the is the smallest significance level at which the null hypothesis could be rejected. Given p-value we do not have to compare test statistics with the corresponding critical value. If the p-value is lower than the significance level (α) then we can reject the null. Jakub Mućk Applied Econometrics (QEM) Meeting #3 25 / 42

35 Linear combination of Parameters I t-test P-value Linear combination of Parameters Normality of the error term A linear combination of parameters: where c 1 and c 2 are some constants. λ = c 1 β 1 + c 2 β 2 (22) Under the assumptions #1-#6 (without normality of the error term) LS LS the least square estimators ˆβ 1 and ˆβ 2 are the best linear unbiased estimators of β 1 and β 2. Moreover, the ˆλ LS = c 1 ˆβLS 1 + c 2 ˆβLS 2 is also BLUE of λ. The estimator ˆλ LS is unbiased because: E(ˆλ LS ) = E(c ˆβLS 1 1 )+E(c ˆβLS LS LS 2 2 ) = c 1E( ˆβ 1 )+c 1E( ˆβ 2 ) = c 1β 1+c 2β 2 = λ. (23) Jakub Mućk Applied Econometrics (QEM) Meeting #3 26 / 42

36 Linear combination of Parameters II t-test P-value Linear combination of Parameters Normality of the error term The variance of the linear combination of the LS estimates: var(ˆλ) = var(c 1 ˆβLS 1 + c 2 ˆβLS = c 1 var( ˆβ LS 1 ) + c 2 var( 2 ) (24) LS LS LS ˆβ 2 ) + 2c 1 c 2 cov( ˆβ 1, ˆβ 2 ). (25) therefore we can estimate the variance of the λ by replacing with the (known) estimated variances and covariance. LS LS LS LS var(ˆλ) ˆ = c 1 var( ˆ ˆβ 1 ) + c 2 var( ˆ ˆβ 2 ) + 2c 1 c 2 cov( ˆ ˆβ 1, ˆβ 2 ) (26) if the assumption of the error term normality holds or if the sample is large then ˆλ have normal distribution: ( ) ˆλ = c ˆβLS c ˆβLS 2 2 N λ, var(ˆλ). (27) Jakub Mućk Applied Econometrics (QEM) Meeting #3 27 / 42

37 t-test P-value Linear combination of Parameters Normality of the error term Linear combination of Parameters III The standard t-statistics for the linear combination is: t = ˆλ λ = ˆλ λ var(ˆλ) se(ˆλ) t N K. (28) Based on the above formulation the variety of hypotheses can be tested. The null is typically: H 0 : λ = c 1 β 1 + c 2 β + 2 = λ 0, (29) while the possible alternative hypotheses: H 0 : λ = c 1 β 1 + c 2 β + 2 λ 0, H 0 : λ = c 1 β 1 + c 2 β + 2 λ 0, H 0 : λ = c 1 β 1 + c 2 β + 2 λ 0, Jakub Mućk Applied Econometrics (QEM) Meeting #3 28 / 42

38 Normality of the error term t-test P-value Linear combination of Parameters Normality of the error term The assumption of the error term is crucial to test the hypothesis. However, the error term is random variable and, therefore, is not unobservable. The normality of the error term can be justified on the basis of the residuals properties. The assessment of this assumption bases on: the residuals histogram, results of the Jarque-Berra test. But if the sample is sufficiently large then, according to a central limit theorem, the distribution of least squares estimator can be approximated by normal distribution. Jakub Mućk Applied Econometrics (QEM) Meeting #3 29 / 42

39 The Jarque-Berra test t-test P-value Linear combination of Parameters Normality of the error term In general, the Jarque-Berra test allows to investigate whether sample data have the skewness and kurtosis that match to normal distribution. The skewness (S) and kurtosis (K) of residuals (ê i ) S = ( 1 N 1 N N (êi ˆē ) 3 i=1 N (êi ˆē ) 2 i=1 ) 3 2 and K = ( 1 N 1 N N (e i ē) 4 i=1 N (êi ˆē ) ) i=1 The test statistics: J B = N 6 (S ) (K 3)2 χ 2 (2). (30) Jakub Mućk Applied Econometrics (QEM) Meeting #3 30 / 42

40 Outline t-test P-value Linear combination of Parameters Normality of the error term 4 5 R 2 Adjusted R 2 Information Criteria Jakub Mućk Applied Econometrics (QEM) Meeting #3 31 / 42

41 Economic variables are not always related by straight-line relationships. They display curvilinear forms. [Example] Wages (w) and experience (exper): w = β 0 + β 1 exper + β 2 exper 2 + ε. (31) In the above model, the quadratic relationship is assumed. Why? In general, the choice of function form is related to: 1 economic theory, 2 empirical pattern, 3 properties of residuals. The most popular nonlinear functions: quadratic and cubic relationship, polynomial equations, logs of the dependent and/or independent variable. Jakub Mućk Applied Econometrics (QEM) Meeting #3 32 / 42

42 Economic variables are not always related by straight-line relationships. They display curvilinear forms. [Example] Wages (w) and experience (exper): w = β 0 + β 1 exper + β 2 exper 2 + ε. (31) In the above model, the quadratic relationship is assumed. Why? In general, the choice of function form is related to: 1 economic theory, 2 empirical pattern, 3 properties of residuals. The most popular nonlinear functions: quadratic and cubic relationship, polynomial equations, logs of the dependent and/or independent variable. Jakub Mućk Applied Econometrics (QEM) Meeting #3 32 / 42

43 Examples Orange line : y = β 1 + β 2 x Red line : y = β 1 + β 2 x Jakub Mućk Applied Econometrics (QEM) Meeting #3 33 / 42

44 Examples Orange line : y = β 1 + β 2 x Red line : y = β 1 + β 2 ln x Jakub Mućk Applied Econometrics (QEM) Meeting #3 33 / 42

45 Examples Orange line : y = β 1 + β 2 x Red line : ln y = β 1 + β 2 x Jakub Mućk Applied Econometrics (QEM) Meeting #3 33 / 42

46 Examples Orange line : y = β 1 + β 2 x Red line : ln y = β 1 + β 2 ln x Jakub Mućk Applied Econometrics (QEM) Meeting #3 33 / 42

47 How to interpret coefficients Marginal effects measures expected instantaneous change in the dependent variable (y)in a reaction to change in explanatory variable (x): Marginal effect = E (y) x (32) In other words, the marginal effects is the slope of the tangent to the curve at a particular point. Elasticity measures the percentage change in y in a reaction to percentage change in x: Elasticity = E (y) x x y. (33) Semi-elasticity measures the percentage change in y in a reaction to a change in x E (y) 1 Semi-Elasticity = x y. (34) Jakub Mućk Applied Econometrics (QEM) Meeting #3 34 / 42

48 Some useful functions Name Function Slope Elasticity (marginal effects) x Linear y = β 0 + β 1 x β 1 β 1 y Quadratic y = β 0 + β 1 x 2 2β 1 x 2β 1 x x y Quadratic (II) y = β 0 + β 1 x + β 2 x 2 β 1 + 2β 2 x (β 1 + 2β 2 x) x y Cubic y = β 0 + β 1 x 3 3β 1 x 2 3β 1 x 2 x y y Log-Log ln(y) = β 0 + β 1 ln(x) β 1 x β 1 Log-Linear ln(y) = β 0 + β 1 x β 1 y β 1 x a 1 unit change in x leads to (approximately) a 100 β 1 % change in y 1 Linear-Log y = β 0 + β 1 ln(x) β 1 x β 1 1y a 1 % change in x leads to (approximately) a β 1 /100 unit change in y Jakub Mućk Applied Econometrics (QEM) Meeting #3 35 / 42

49 Interaction variable Interaction variable is the product of (at least) two variable involved in regression and accounts for simultaneous effects of two variables. [Example] Wages (w), experience (exper) and education (educ): w = β 0 + β 1 exper + β 2 exper 2 + β 3 educ + β 4 exper educ + ε. (35) In this case: Marginal effect of education = Marginal effect of education = E (w) educ = β 3 + β 4 exper, E (w) exper = β 2 + 2β 3 exper + β 4 educ. Jakub Mućk Applied Econometrics (QEM) Meeting #3 36 / 42

50 Outline R 2 Adjusted R 2 Information Criteria t-test P-value Linear combination of Parameters Normality of the error term 4 5 R 2 Adjusted R 2 Information Criteria Jakub Mućk Applied Econometrics (QEM) Meeting #3 37 / 42

51 I R 2 Adjusted R 2 Information Criteria The observed values (y i ) of dependent variable can be decomposed into the fitted values (ŷ i ) and the residuals (ê i ): y i = ŷ i + ê i, (36) subtracting the sample mean (ȳ) from both sides: y i ȳ = ŷ i ȳ + ê i. (37) Squaring and summing both sides of above equation: N (y i ȳ) 2 = i=1 N N (ŷ i ȳ) 2 + êi 2, (38) i=1 In the above expression we use assumption that N i=1 (ŷ i ȳ) ê i = 0 since the x 1,..., x K are not random. i=1 Jakub Mućk Applied Econometrics (QEM) Meeting #3 38 / 42

52 II R 2 Adjusted R 2 Information Criteria The decomposition of total variation in dependent variable: SST = SSR + SSE, (39) where SST is the sum of squares and SST = N i=1 (yi ȳ)2, SSR is the sum of squares due to regression and SSR = N i=1 (ŷi ȳ)2, SSE is the sum of squares due to regression and SSE = N i=1 ê2 i. Coefficient of determination R 2 is the proportion of variation that can be explained by independent variables: R 2 < 0, 1 >. R 2 = SSR SST = 1 SSE SST, (40) Jakub Mućk Applied Econometrics (QEM) Meeting #3 39 / 42

53 Correlation coefficient and R 2 R 2 Adjusted R 2 Information Criteria The correlation coefficient ρ xy between x and y is defined by: ρ xy = and the sample correlation coefficient takes the values between 1 and 1. cov(x, y) var(x) var(y) = σ xy σ x σ y, (41) r xy = s xy s x s y, (42) In simple linear regression: the relationship between R 2 and r xy is as follows: R 2 = r 2 xy, (43) and, therefore, the R 2 can also be computed as the square of the sample correlation coefficient between y i and ŷ i = ˆβ 0 + ˆβ 1 x i. Jakub Mućk Applied Econometrics (QEM) Meeting #3 40 / 42

54 Adjusted R 2 R 2 Adjusted R 2 Information Criteria The coefficient of determination R 2 is always higher if we include additional explanatory variable even if the added variable is not justified/ relevant. The adjusted coefficient of determination R 2 : R 2 = 1 SSE (N 1) SST (N K), (44) where SSE is the sum of squared errors and SST is the sum of squares. With the adjusted coefficient of determination we account for a decrease in degree of freedoms: (N 1)/(N K). However, it has no convenient interpretation. Jakub Mućk Applied Econometrics (QEM) Meeting #3 41 / 42

55 Information Criteria R 2 Adjusted R 2 Information Criteria Information criteria are alternative measures of goodness-of-fit. They have no interpretation but, like adjusted R 2, account for a decrease in degrees of freedom. The Akaike information criterion (AIC): ( ) SSE AIC = ln + 2K N N. (45) The Bayesian information criterion (SIC): ( ) SSE SIC = ln + K ln(k). (46) N N Using the above criteria, the lower values of AIC/BIC signals better fit to data. Jakub Mućk Applied Econometrics (QEM) Meeting #3 42 / 42