ECON301_Handout_08_1213_02

# 1 before moving on we want to discuss a bit further

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1 Before moving on, we want to discuss a bit further the interpretation of equation (1). Given the values of the regressor, the random variable in the probability statement is ˆ i . Equation (1) says that the probability is 0.95 that the interval /2 ˆ /2 ˆ ˆ ˆ ˆ ˆ ( . , . ) i i i i t t will contain i . In other words, assume that we could take a large number of samples, say 1000 of size 50 each. Assume, moreover, that for each of these samples the values of the t X ’s are the same. Suppose that we calculate ˆ i for each sample. Since the disturbance terms would be expected to differ from sample to sample, the values of the t Y ’s would differ among the samples even though the set of values of the t X ’s remain unchanged. Since ˆ i depends on the t Y ’s, it wou ld vary across samples. If we then were to calculate the interval 1 Put differently, the larger is the standard error of the estimator, the greater the uncertainty of estimating the true value of the unknown parameter. Thus, the standard error of an estimator is often described as a measure of the precision of the estimator: how precisely the estimator measures the true population value.

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ECON 301 - Introduction to Econometrics I April, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: Lecture Notes 5 /2 ˆ /2 ˆ ˆ ˆ ˆ ˆ ( . , . ) i i i i t t for each sample, we would essentially have 1000 different intervals. The essence of equation (1) is that with this procedure we would expect 0.95(1000)=950 of these intervals to contain the constant i . Note that, in matrix terms, the estimated variance of the sampling distributions is as follows: 2 1 ˆ ˆ var( ) ( ) β XX o For a particular β j , just pick off the main diagonal element of this matrix, and take its square root to get the standard error. Example 1 Suppose we have estimated the following regression line from a sample of 13 observations where 1 ˆ ( ) se =0.07 (0.07) ˆ 8.5 0.724 t t Y X a) Test 1 0.570 at 0.05 level.
ECON 301 - Introduction to Econometrics I April, 2012

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