Economics 326 Methods of Empirical Research in Economics. Lecture 7: Con dence intervals

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1 Economics 326 Methods of Empirical Research in Economics Lecture 7: Con dence intervals Hiro Kasahara University of British Columbia December 24, 2014

2 Point estimation I Our model: 1 Y i = β 0 + β 1 X i + U i, i = 1,, n 2 E (U i jx 1,, X n ) = 0 for all i s 3 E U 2 i jx 1,, X n = σ 2 for all i s 4 E U i U j jx 1,, X n = 0 for all i 6= j 5 U s are jointly normally distributed conditional on X s I The OLS estimator is a point estimator of β 1 I For our model, conditional on X s: N β 1, Var, Var = σ 2 n i=1 (X i X ) 2 I With probability one, we have that 6= β 1 1/12

3 Interval estimation problem I We want to construct an interval estimator for β 1 : I The interval estimator is called a con dence interval (CI) I A CI contains the true value β 1 with some pre-speci ed probability 1 α, where α is a small probability of error I For example, if α = 005, then the random CI will contain β 1 with probability 095 I 1 α is called the coverage probability I Con dence interval: CI 1 α = [LB 1 α, UB 1 α ] The lower bound (LB) and upper bound (UB) should depend on the coverage probability 1 α I The formal de nition of CI: It is a random interval CI 1 that conditional on X s, P (β 1 2 CI 1 α ) = 1 α Note that the random element is CI 1 α I Sometimes, a CI is de ned as P (β 1 2 CI 1 α ) 1 α α such 2/12

4 Symmetric CIs I One approach to constructing CIs is to consider a symmetric interval around the estimator : CI 1 α = c 1 α, + c 1 α I The problem is choosing c 1 α such that P (β 1 2 CI 1 α ) = 1 α I In choosing c 1 α we will be relying on the fact that given our assumptions and conditionally on X s: N β 1, Var and Var = σ 2 I Note that conditionally on X s: β 1 N (0, 1) Var n i=1 (X i X ) 2 3/12

5 Quantiles (percentiles) of the standard normal distribution I Let Z N (0, 1) The τ-th uantile (percentile) of the standard normal distribution is z τ such that P (Z z τ ) = τ I Median: τ = 05 and z 05 = 0 (P (Z 0) = 05) I If τ = 0975 then z 0975 = 196 Due to symmetry, if τ = 0025 then z 0025 = 196 4/12

6 σ 2 is known (infeasible CIs) I Suppose (for a moment) that σ 2 is known, and we can compute exactly the variance of as Var = σ 2 / n i=1 (X i X ) 2 I Consider the following CI: CI 1 α = z 1 α/2 Var, + z 1 α/2 Var I For example, if 1 α = 095 () α = 005 () z 1 α/2 = z 0975 = 196, and CI 095 = 196 Var, Var 5/12

7 Validity of the infeasible CIs (σ 2 is known) I We need to show that P β 1 2 z 1 α/2 Var, + z 1 α/2 Var = 1 α I Next, z 1 α/2 Var β1 + z 1 α/2 Var () z 1 α/2 Var β1 z 1 α/2 Var () z 1 α/2 Var β 1 z 1 α/2 Var () z 1 α/2 β 1 Var ˆβ z 1 α/2 1 6/12

8 Validity of the infeasible CIs (σ 2 is known) I We have that β 1 2 z 1 α/2 Var, + z 1 α/2 Var () z 1 α/2 β 1 Var ˆβ z 1 α/2 1 I Let Z = β 1 N (0, 1) Var( ) 0 1 z 1 α/2 β 1 Var ˆβ z A 1 α/2 1 = P ( z 1 α/2 Z z 1 α/2 ) = P (z α/2 Z z 1 α/2 ) = 1 α/2 α/2 = 1 α 7/12

9 Feasible con dence intervals (σ 2 is unknown) I Since σ 2 is unknown, we must estimate it from the data: s 2 = 1 n 2 n Ûi 2 = 1 i=1 n 2 n i=1 Y i ˆβ 0 X i 2 I We can replace σ 2 by s 2, however, the result does not have a normal distribution any more: β 1 d Var t n 2, where dvar = s 2 n i=1 (X i X ) 2 Here t n 2 denotes the t-distribution with n 2 degrees of freedom I The degrees of freedom depend on I I the sample size (n), and the number of parameters one have to estimate to compute s 2 (two in this case, β 0 and β 1 ) 8/12

10 Feasible con dence intervals (σ 2 is unknown) I Let t df,τ be the τ-th uantile of the t-distribution with the number of degrees of freedom df : If T t df then P (T t df,τ ) = τ I Similarly to the normal distribution, the t-distribution is centered at zero and is symmetric around zero: t n 2,1 α/2 = t n 2,α/2 I We can now construct a feasible con dence interval with 1 α coverage as: = CI 1 α = t n 2,1 α/2 dvar, + t n 2,1 α/2 dvar, where dvar = s 2 n i=1 (X i X ) 2 9/12

11 Example: Rent rates and average income I Data (RENTALDTA): 64 cities in 1990, Rent = average rent, AvgInc = per capita income: Rent i = β 0 + β 1 AvgInc i + U i I t 62, =)The 95% con dence interval for β 1 is [ , ] = [00089, 00141] I t 62, =)The 90% con dence interval for β 1 is [ , ] = [00093, 00137] 10/12

12 The e ect of estimation of σ 2 I The t-distribution has heavier tails than normal The graphs of normal (solid line), t 5 (dashed line), and t 1 (dotted line) PDFs: I t df,1 α/2 > z 1 α/2, but as df increases t df,1 α/2! z 1 α/2 I When the sample size n is large, t n 2,1 α/2 can be replaced with z 1 α/2 11/12

13 Interpretation of con dence intervals I The con dence interval CI 1 α is a function of the sample f(y i, X i ) : i = 1,, ng, and therefore is random This allows us to talk about probability of CI 1 α containing the true value of β 1 I Once the con dence interval is computed given the data, we have its one realization The realization of CI 1 α or (computed con dence interval) is not random, and it does not make sense anymore to talk about the probability that it includes the true β 1 I Once the con dence interval is computed, it either contains the true value β 1 or it does not 12/12