Principles of Econometrics 4th Edition Page 71 Chapter 3 Interval Estimation

Principles of econometrics 4th edition page 71

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Principles of Econometrics, 4th Edition Page 71 Chapter 3:  Interval Estimation and Hypothesis Testing The variance of is where the variances and covariances are given in Eq. 2.20 - 2.22 ˆ 2 1 2 1 2 2 2 1 2 1 2 2 1 1 , cov 2 var var var ˆ var b b c c b c b c b c b c Eq. 3.8 3.6 Linear  Combinations of  Parameters
Principles of Econometrics, 4th Edition Page 72 Chapter 3:  Interval Estimation and Hypothesis Testing We estimate by replacing the unknown variances and covariances with their estimated variances and covariances in Eq. 2.20 - 2.22 ˆ 1 1 2 2 2 2 1 1 2 2 1 2 1 2 ˆ var var var var 2 cov , c b c b c b c b c c b b Eq. 3.9 3.6 Linear  Combinations of  Parameters
Principles of Econometrics, 4th Edition Page 73 Chapter 3:  Interval Estimation and Hypothesis Testing The standard error of is the square root of the estimated variance ˆ 1 1 2 2 1 1 2 2 ˆ se se var c b c b c b c b Eq. 3.10 3.6 Linear  Combinations of  Parameters
Principles of Econometrics, 4th Edition Page 74 Chapter 3:  Interval Estimation and Hypothesis Testing If in addition SR6 holds, or if the sample is large, the least squares estimators b 1 and b 2 have normal distributions. It is also true that linear combinations of normally distributed variables are normally distributed, so that ˆ var , ~ ˆ 2 2 1 1 N b c b c 3.6 Linear  Combinations of  Parameters
Principles of Econometrics, 4th Edition Page 75 Chapter 3:  Interval Estimation and Hypothesis Testing We can estimate the average (or expected) expenditure on food as: If the household income is \$2000, which is 20 since income is measured in \$100 units in this example, then the average expenditure is: We estimate that the expected food expenditure by a household with \$2,000 income is \$287.61 _ 83.4160 10.2096 FOOD EXP INCOME 3.6.1 Estimating  Expected Food  Expenditure 1 2 _ | 20 20 83.4160 10.2096 20 287.6089 E FOOD EXP INCOME b b 3.6 Linear  Combinations of  Parameters
Principles of Econometrics, 4th Edition Page 76 Chapter 3:  Interval Estimation and Hypothesis Testing The t-statistic for the linear combination is: 3.6.2 An Interval  Estimate of  Expected Food  Expenditure 1 1 2 2 1 1 2 2 1 1 2 2 2 ˆ ˆ var ˆ ˆ se β β se ~ N t c b c b c c c b c b t Eq. 3.11 3.6 Linear  Combinations of  Parameters
Principles of Econometrics, 4th Edition Page 77 Chapter 3:  Interval Estimation and Hypothesis Testing Substituting the t value into P (- tc t tc ) = 1 – α, we get: so that the (1 – α)% interval is 3.6.2 An Interval  Estimate of  Expected Food  Expenditure 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 seβ β se c c c b c b t c b c b c c P c b c b t c b c b   1 1 2 2 1 1 2 2 se c c b c b t c b c b 3.6 Linear  Combinations of  Parameters
Principles of Econometrics, 4th Edition Page 78 Chapter 3:  Interval Estimation and Hypothesis Testing For our example, the estimated variances and covariance is: 3.6.2 An Interval  Estimate of  Expected Food  Expenditure C Income C 1884.442 -85.9032 Income -85.9032 4.3818 3.6 Linear  Combinations of  Parameters