3.3 Confidence intervals for regression coefficients Confidence intervals (interval estimation): Specifying the region within which the "true" regression coefficient will lie with a large probability Construction of a confidence interval for the unknown regression coefficients β j To this end, we start with the t-value of the regression coefficient (standardized regression coefficient): j ˆ jj j j j j j x ˆ ˆ ˆ Var ˆ t (3.15) which is t-distributed with n-k degrees of freedom. The probability that the standardized regression coefficient lies within a symmetric interval around its expected value 0, is given by 1 t t t P 2 / 1 ; k n 2 / 1 ; k n With (3.15), we obtain t nk; 1-α/ 2 : (1-α/ 2)-quantile of the t-distribution with n-k degrees of freedom
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3.3 Confidence intervals for regression coefficients · Excursion: Chi-square and F- -distribution- Chi-Square distribution-Let Z1,Z2,K,Zn, σ Xi μ Zi , be n independently, normally
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3.3 Confidence intervals for regression coefficients
Confidence intervals (interval estimation):
Specifying the region within which the "true" regression coefficient will lie with
a large probability
Construction of a confidence interval for the
unknown regression coefficients βj
To this end, we start with the t-value of the regression coefficient
(standardized regression coefficient):j
jj
jj
j
jj
xˆ
ˆ
ˆVar
ˆt
(3.15)
which is t-distributed with n-k degrees of freedom. The probability that the
standardized regression coefficient lies within a symmetric interval around
its expected value 0, is given by
1tttP 2/1;kn2/1;kn
With (3.15), we obtain
tnk; 1-α/ 2: (1-α/ 2)-quantile of the t-distribution with n-k degrees of freedom
1t
xˆ
ˆtP 2/1;kn
jj
jj2/1;kn
and, after a further transformation
1xˆtˆxˆtˆP jj2/1;knjj
jj2/1;knj(3.16)
Equation (3.16) indicates the confidence interval for βj at a confidence level
(= confidence probability) of 1-α. In the case of a normally distributed error
term u, the unknown regression coefficient βj lies with a probability of 1-α within
the interval
jj2/1;knj xˆtˆ
Confidence intervals for β1 and β2 in the simple regression:
2
t2t
2t
2/1;2n1xxn
xˆtˆ
2
t2t
2/1;2n2xxn
nˆtˆ
and
Example:
As a point estimate for the marginal propensity to consume c1 of the macro-
economic consumption function the value 0,9335 has been determined. For
economic policy measures it is however also of interest to know, at a high level
of confidence, the range within which the marginal propensity to consume will
likely lie. For this reason the point estimate for the marginal propensity to consume
is to be supplemented with a confidence interval.
A 95% confidence interval for the marginal propensity is given by
2
t2t
975,0;2n1xxn
nˆtc
The numerical values that are required in order to calculate the standard
deviation of c1 are already known:
9,26529,37472832,19,1103,8ˆ 2
tt xxn
Using the 0.975-quantile t17; 0.975 = 2.110 of the t-distribution with
17 degrees of freedom one obtains the confidence interval
for the marginal propensity to consume at the 95% confidence level. □