Econometrics. 4) Statistical inference

30C00200 Econometrics 4) Statistical inference Timo Kuosmanen Professor, Ph.D. http://nomepre.net/index.php/timokuosmanen

Today s topics Confidence intervals of parameter estimates Student s t-distribution Hypothesis testing t-test of significance of coefficients p-value One-sided vs. two-sided tests Type I and II errors in hypothesis testing Power of a test

Types of statistical inference Estimation Point estimation Interval estimation Hypothesis testing

SUMMARY OUTPUT Excel output of the hedonic model - Multiple regression Regression Statistics Multiple R 0,905971 R Square 0,820784 Adjusted R Square 0,81225 Standard Error 80593,26 Observations 67 ANOVA df SS MS F Significance F Regression 3 1,87E+12 6,25E+11 96,17703 1,76E-23 Residual 63 4,09E+11 6,5E+09 Total 66 2,28E+12 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 141366,7 36401,58 3,883532 0,000249 68623,96 214109,5 size m2 6972,404 857,5745 8,130378 2,11E-11 5258,678 8686,13 nr. bedrooms -65182,6 20401,6-3,19498 0,002186-105952 -24413,3 age -2820,1 495,0578-5,6965 3,46E-07-3809,39-1830,8

Interval estimation Definition: Let X be a random sample from a probability distribution with parameter μ. The 100 (1-α)% confidence interval for parameter μ is an interval with random endpoints [a(x), b(x)] determined by random sample X, such that Pr( a( X ) b( X )) 1 Interpretation in terms of repeated samples: suppose we draw a large number of random samples X from the population, and calculate the confidence interval for each sample. The calculated confidence interval (which would differ for each sample) would encompass the true population parameter μ in 100 (1-α)% of samples. 5

Interval estimation Definition: Let X be a random sample from a probability distribution with parameter μ. The 100 (1-α)% confidence interval for parameter μ is an interval with random endpoints [a(x), b(x)] determined by random sample X, such that Pr( a( X ) b( X )) 1 Significance α is usually specified as 5% or 1% For a given sample X and significance α, our objective is to calculate the lower limit a the upper limit b. 6

Asymptotic normality By the central limit theorem, if the assumptions Exogeneity, Homoscedasticity, and Serial independence hold, then b 2 converges in distribution to the normal distribution b 2 ~ N, ( n 1) Var ( x ) 2 2 a Denoting 2 b2 2 ( n 1) Var( x) We have a shorter expression: b ~ N, b 2 2 a 2 2

Standardization The result b ~ N, b 2 2 a 2 2 implies that ( b )~ N 0, b 2 2 2 a 2 and further ( b2 2) ~ N 0,1 a b2

Standard normal distribution Ф is the cumulative density function of N(0,1) Ф(-1.96)=0.025 Ф(1.96)=0.975 Ф(-2.56)=0.005 Ф(2.56)=0.995

95% confidence interval for slope β 2 Using ( b2 2) ~ N 0,1 a b2 and Ф(-1.96)=0.025, Ф(1.96)=0.975 when n is sufficiently large ( b ) b2 2 2 Pr 1.96 0.025 ( b ) b2 2 2 Pr 1.96 0.975 Thus, ( b ) 2 2 Pr 1.96 1.96 0.95 b2

95% confidence interval for slope β 2 Modifying ( b ) 2 2 Pr 1.96 1.96 0.95 b2 Pr 1.96 b 1.96 0.95 b2 2 2 b2 Pr b 1.96 b 1.96 0.95 2 b2 2 2 b2 Thus, the 95% confidence interval takes the form b 1.96 b 2 2

Confidence interval for slope β 2 Recall the 95% confidence interval b 1.96 b 2 2 If n is very large, we can substitute the true but unknown standard deviation σ b2 by the estimated standard error However, in small samples the estimation of σ b2 causes an additional source of variation that should be taken into account in the confidence interval -> we need to take the critical value from the Student s t distribution rather than from N(0,1) (i.e., 1.96 used above) b t s. e.( b ) 12 2 crit 2

Student s t distribution t distribution depends on the degrees of freedom (df): it converges to N(0,1) as df increases For OLS estimator, df = n - K where K = number of unknown model parameters (β s) 13

Student s t distribution Confidence interval b t s. e.( b ) 2 crit 2 Example of critical t values at 5% significance level with different sample sizes (df= n-2) Excel function =TINV(prob; df) n t crit 20 2.101 50 2.011 100 1.984 200 1.972 500 1.965 1,000 1.962 2,000 1.961 5,000 1.960 14

Classic approach: Hypothesis testing 1) State the null hypothesis (H 0 ) and the alternative hypothesis (H 1 ). 2) Specify the probability model under H 0 and the necessary assumptions. 3) Compute the test statistic (S) with a known probability distribution under H 0. 4) Identify the acceptance and rejection regions, given the known probability distribution of S and the pre-assigned significance level. 5) Accept H 0 if S falls within the acceptance region; Reject H 0 if S falls within the rejection region. 15

Testing hypotheses concerning β s Three alternative approaches: 1) Confidence intervals 2) t-test 3) p-value 16

Tests of β s using confidence intervals We can use confidence intervals for testing hypotheses Applies to all two-sided tests Significance test: H 0 : β 2 = 0 (x has no effect on y) H 1 : β 2 0 Tests of theoretical restrictions: H 0 : β 2 = β* (based on theory) H 1 : β 2 β* 17

Tests of β s using confidence intervals At the given significance level α: Estimate the 100%(1- α) confidence interval and state the hypotheses to be tested. H 0 : β 2 = β*, H 1 : β 2 β* Accept H 0 if β* is contained within the 100%(1- α) confidence interval Reject H 0 if β* falls outside the 100%(1- α) confidence interval 18

t-test To derive the test statistic, recall that b 2 2 b2 ~ N 0,1 a Further, using the standard error estimated from data, we have b2 2 ~ tn ( 2) s. e.( b ) 2 20

t-test H 0 : β 2 = β* H 1 : β 2 β* If H 0 is true, then we can use the test statistic (t stat): t b 2 * s. e.( b ) 2 If H 0 is true, then our test statistic follows Student s t distribution with (n-2) degrees of freedom. 21

t-test H 0 : β 2 = β* H 1 : β 2 β* Acceptance region: If -t crit < t < t crit, then maintain H 0 Rejection region: t < -t crit or t > t crit, then reject H 0 22

Significance test In the case of the significance test: H 0 : β 2 = 0, H 1 : β 2 0 Test statistic: t b 2 s. e.( b ) 2 The value of this test statistic reported as a part of the Stata output. The reported value should be compared with t crit obtained from statistical tables (or e.g. Excel) 23

p-value From reported t-statistics, it is not always directly obvious whether coefficient is statistically significant at 5%, 1%, or perhaps 10% significance levels. Need to compare with t crit that depend on n and α The p- value indicates directly the probability of obtaining the observed t or higher when H 0 is true. The probability of Type I error when H 0 is true. The p- value indicates directly the smallest significance level α at which H 0 can be rejected. 25

One-sided vs. two-sided test Two-sided test H 0 : β 2 = β* H 1 : β 2 β* One-sided tests H 0 : β 2 = β* H 1 : β 2 < β* [or H 1 : β 2 > β*] The sign or direction of the deviation from the null hypothesis is known from theory or experience. 26

One-sided vs. two-sided test Two-sided test H 0 : β 2 = β* H 1 : β 2 β* One-sided tests H 0 : β 2 = β* H 1 : β 2 < β* 27

One-sided vs. two-sided test Example: critical t values at 5% significance levels, df = n-2 one-sided two-sided 20 1,734 2,101 50 1,677 2,011 100 1,661 1,984 200 1,653 1,972 500 1,648 1,965 1000 1,646 1,962 2000 1,646 1,961 5000 1,645 1,960 28

One-sided vs. two-sided test Impacts of one-sided testing: Decrease in the critical t value Easier to reject H 0 Increases both the size and the power of the test 29

Comparison of the 3 approaches Confidence intervals + applies to both significance tests and theoretical restrictions - two-sided tests only - fixed significance level (= 1 confidence level) p-value - significance test - two-sided tests only + any significance level can be used t-statistic - significance test (but t-stats for theoretical restrictions can be computed) + applies to both one-sided and two-sided tests + any significance level can be used - need to find critical value of the t-stat from statistical tables (or Excel) 30

Interpretation of the test Important: if H 0 is accepted, it does not mean that H 0 has been proved to be true. The null hypothesis is assumed to be true from the start of the test; if there is not enough evidence to reject the null, it simply continues to be assumed true. Statistical test can fail to reject H 0 even when H 0 is false Statistical power of the test! 31

Two possible types of error Accept H 0 Reject H 0 H 0 is true Correct Type I error H 0 is false Type II error Correct 32

Size of a test The probability of a type I error is called the size of the test. This is directly controlled for by setting the significance level α. Setting α = 5% means that we tolerate 5% risk of rejecting H 0 when it is in fact true. 33

Power of a test The power of the test is the probability that it will correctly lead to rejection of a false null hypothesis: power = 1 Prob(type II error) = 1 - β For a given α, we would like β to be as small as possible Most powerful test Tradeoff: decreasing the probability of type I error, probability of type II error increases, and vice versa. 34

Topic: Next time Mon 21 Sept Dummy variables 35