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Confidence intervals were treated at length in the Review chapter and their application to regression analysis presents no problems. We will not repeat the graphical explanation. We will just provide the mathematical derivation in the context of a regression.
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CONFIDENCE INTERVALS FOR REGRESSION COEFFICIENTS
Model
Null hypothesis:
Alternative hypothesis:
0220 : H0221 : H
uXY 21
From the initial discussion in this section, we saw that, given the theoretical model Y = 1 + 2X + u and a fitted model, the regression coefficient b2 and the hypothetical value of 2 are incompatible if either of the inequalities shown is valid.
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CONFIDENCE INTERVALS FOR REGRESSION COEFFICIENTS
crit2
022
s.e.t
bb
crit2
022
s.e.t
bb
Model
Null hypothesis:
Alternative hypothesis:
0220 : H0221 : H
uXY 21
Reject H0 if or
Multiplying through by the standard error of b2, the conditions for rejecting H0 can be written as shown.
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CONFIDENCE INTERVALS FOR REGRESSION COEFFICIENTS
crit2
022
s.e.t
bb
crit2
022
s.e.t
bb
crit2022 s.e. tbb crit2
022 s.e. tbb
Model
Null hypothesis:
Alternative hypothesis:
0220 : H0221 : H
uXY 21
Reject H0 if or
Reject H0 if or
The inequalities may then be re-arranged as shown.
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CONFIDENCE INTERVALS FOR REGRESSION COEFFICIENTS
crit2
022
s.e.t
bb
crit2
022
s.e.t
bb
crit2022 s.e. tbb crit2
022 s.e. tbb
02crit22 s.e. tbb 0
2crit22 s.e. tbb
Model
Null hypothesis:
Alternative hypothesis:
0220 : H0221 : H
uXY 21
Reject H0 if or
Reject H0 if or
Reject H0 if or
We can then obtain the confidence interval for 2, being the set of all values that would not be rejected, given the sample estimate b2. To make it operational, we need to select a significance level and determine the corresponding critical value of t.
crit2
022
s.e.t
bb
crit2
022
s.e.t
bb
crit2022 s.e. tbb crit2
022 s.e. tbb
02crit22 s.e. tbb 0
2crit22 s.e. tbb
crit222crit22 s.e.s.e. tbbtbb
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Model
Null hypothesis:
Alternative hypothesis:
CONFIDENCE INTERVALS FOR REGRESSION COEFFICIENTS
0220 : H0221 : H
uXY 21
Reject H0 if or
Reject H0 if or
Reject H0 if or
Do not reject H0 if
For an example of the construction of a confidence interval, we will return to the earnings function fitted earlier. We will construct a 95% confidence interval for 2.
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CONFIDENCE INTERVALS FOR REGRESSION COEFFICIENTS
. reg EARNINGS S
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 1, 538) = 112.15 Model | 19321.5589 1 19321.5589 Prob > F = 0.0000 Residual | 92688.6722 538 172.283777 R-squared = 0.1725-------------+------------------------------ Adj R-squared = 0.1710 Total | 112010.231 539 207.811189 Root MSE = 13.126
------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- S | 2.455321 .2318512 10.59 0.000 1.999876 2.910765 _cons | -13.93347 3.219851 -4.33 0.000 -20.25849 -7.608444------------------------------------------------------------------------------
crit222crit22 s.e.s.e. tbbtbb
The point estimate 2 is 2.455 and its standard error is 0.232.
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CONFIDENCE INTERVALS FOR REGRESSION COEFFICIENTS
. reg EARNINGS S
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 1, 538) = 112.15 Model | 19321.5589 1 19321.5589 Prob > F = 0.0000 Residual | 92688.6722 538 172.283777 R-squared = 0.1725-------------+------------------------------ Adj R-squared = 0.1710 Total | 112010.231 539 207.811189 Root MSE = 13.126
------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- S | 2.455321 .2318512 10.59 0.000 1.999876 2.910765 _cons | -13.93347 3.219851 -4.33 0.000 -20.25849 -7.608444------------------------------------------------------------------------------
crit222crit22 s.e.s.e. tbbtbb 2.455 – 0.232 x 1.965 ≤ 2 ≤ 2.455 + 0.232 x 1.965
The critical value of t at the 5% significance level with 538 degrees of freedom is 1.965.
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CONFIDENCE INTERVALS FOR REGRESSION COEFFICIENTS
. reg EARNINGS S
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 1, 538) = 112.15 Model | 19321.5589 1 19321.5589 Prob > F = 0.0000 Residual | 92688.6722 538 172.283777 R-squared = 0.1725-------------+------------------------------ Adj R-squared = 0.1710 Total | 112010.231 539 207.811189 Root MSE = 13.126
------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- S | 2.455321 .2318512 10.59 0.000 1.999876 2.910765 _cons | -13.93347 3.219851 -4.33 0.000 -20.25849 -7.608444------------------------------------------------------------------------------
crit222crit22 s.e.s.e. tbbtbb 2.455 – 0.232 x 1.965 ≤ 2 ≤ 2.455 + 0.232 x 1.965
. reg EARNINGS S
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 1, 538) = 112.15 Model | 19321.5589 1 19321.5589 Prob > F = 0.0000 Residual | 92688.6722 538 172.283777 R-squared = 0.1725-------------+------------------------------ Adj R-squared = 0.1710 Total | 112010.231 539 207.811189 Root MSE = 13.126
------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- S | 2.455321 .2318512 10.59 0.000 1.999876 2.910765 _cons | -13.93347 3.219851 -4.33 0.000 -20.25849 -7.608444------------------------------------------------------------------------------
Hence we establish that the confidence interval is from 1.999 to 2.911. Stata actually computes the 95% confidence interval as part of its default output, 2.000 to 2.911. The discrepancy in the lower limit is due to rounding error in the calculations we have made.
crit222crit22 s.e.s.e. tbbtbb
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CONFIDENCE INTERVALS FOR REGRESSION COEFFICIENTS
2.455 – 0.232 x 1.965 ≤ 2 ≤ 2.455 + 0.232 x 1.965
1.999 ≤ 2 ≤ 2.911
Copyright Christopher Dougherty 2012.
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Introduction to Econometrics, fourth edition 2011, Oxford University Press.
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2012.10.28
