of Properties Estimators
Statistical properties of a and ^ ^ b Mean and variance of ^ b
E(^ = b b)
2 ^ = var(b) SXX
Recall that
n
SXX =
i=1
(xi - x)2
Distribution of ^ b
2 ^ N b, b SXX
Mean and variance of a ^
E(^) = a a
var(^) = a x2 1 + 2 n SXX Distribution of a ^ 1 x2 a N a, ^ + 2 n SXX
Inference for Regression, Mar 1, 2004
-1-
Confidence Intervals
Note that ^ N b, S . Thus b XX ^-b b N (0, 1) / SXX Substituting se for , we obtain ^-b b tn-2 se / SXX (1 - ) confidence interval for b: ^ tn-2,a/2 se b SXX
2
Similarly a-a ^
1 n X2 SXX
N (0, 1)
+
Substituting se for , we obtain a-a ^ se
1 n x2 SXX
tn-2
+
(1 - ) confidence interval for a: a tn-2,/2 se ^
1 x2 + n SXX
Inference for Regression, Mar 1, 2004
-2-
Tests on the Coefficients
Question: Is b equal to some value b0 ? The correspoding test problem is H0 : b = b0 versus Ha : b = b0 .
The test statistic is given by Tb = ^ - b0 b tn-2 se / SXX
The null hypothesis H0 : b = b0 is rejected if |T | > tn-2,/2
Question: Is a equal to some value a0 ? The correspoding test problem is H 0 : a = a0 versus Ha : a = a0 .
The test statistic is given by Ta = se a - a0 ^
1 n x2 SXX
tn-2
+
The null hypothesis H0 : a = a0 is rejected if |T | > tn-2,/2
Inference for Regression, Mar 1, 2004
-3-
Inference for the Coefficients
Example: Body density The confidence interval for b is given by ^ tn-2,/2 se b SXX
0.0132 = [-12.92, -9.90] = -11.41 1.99 0.023
The confidence interval for a is given by a tn-2,/2 se ^ x2 1 + n SXX 0.0132 1 1.062 + = [12.11, 15.30] 92 0.023
= 13.71 1.99 Furthermore we find for
^ b Tb = = -15.22 > t90,0.025 = 1.99 se / SXX Thus we reject H0 : b = 0 at significance level 0.05: The coefficient b is statistically significantly different from zero. Similarly Ta = se a ^
1 n x2 SXX
= 17.26 > t90,0.025 = 1.99
+
Thus we reject H0 : a = 0 at significance level 0.05: The coefficient a is statistically significantly different from zero. The corresponding P -values are P(|Ta | 15.22) 0
P(|Tb| 17.26) 0
Inference for Regression, Mar 1, 2004
-4-
Estimating the Mean
In the linear regression model, the mean of Y at x = x0 is given by
E(Y ) = a + b x0
Our estimate for the mean of Y at X = x0 is ^ Yx0 = a + ^ x0 . ^ b
Question: How precise is this estimate? Note that ^ b(x Yx0 = a + ^ x0 = Y - ^ 0 - x). ^ b Hence we obtain ^ E(Yx ) = a + b x0
0
^ var(Yx0 ) =
1 (x0 - x)2 2 + n SXX
(1 - ) confidence interval for (^ + ^ x0 tn-2,/2 ) se a b
E(Yx )
0
1 (x0 - x)2 + n SXX
Inference for Regression, Mar 1, 2004
-5-
Estimating the Mean
Example: Body density Suppose the measured skin thickness is x0 = 1.1 mm. What is the mean body density for this value of skin thickness? Point estimate: ^ Yx0 = a + hb x0 = 13.71 - 11.41 1.1 = 1.159 ^ The mean body density is 1.159 103 kg/m3 . Confidence interval: (^ + ^ x0 ) tn-2,/2 se a b 1 (x0 - x)2 + n SXX 0.0132
(1.1 - 1.06)2 1 + 92 0.023
= (13.71 - 11.41 1.1) 1.99 = [1.09, 1.22]
In STATA, the standard error for estimating the mean of Y is calculated by passing the option stdp to predict:
. . . . . . > predict BDH predict SE, stdp generate low=BDH-invttail(49,.025)*SE generate high=BDH+invttail(49,.025)*SE sort SKINT graph twoway line low high BDH SKINT, clpattern(dash dash solid) clcolor(black bla ck black) || scatter BODYD SKINT, legend(off) scheme(s1color)
1 1.02
1.2
1.4
1.6
1.8
2
1.04
1.06 SKINT
1.08
1.1
Inference for Regression, Mar 1, 2004
-6-
Prediction
Suppose we want to predict Y at x = x0 . Aim: (1 - ) confidence interval for Y Note that 1 (x0 - X)2 a + ^ x0 - Y N 0, 2 1 + + ^ b n SXX Thus the desired (1 - ) confidence interval for Yx0 is given by a + ^ x0 tn-2,/2 se ^ b 1 (x0 - X)2 1+ + n SXX
Inference for Regression, Mar 1, 2004
-7-
Prediction
Example: Body density Suppose the measured skin thickness is x0 = 1.1 mm. What is the predicted body density for this value of skin thickness? ^ Point estimate: Yx0...