ECON
ECON301_Handout_06_1213_02

# A critical t value is the value that distinguishes

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A critical t -value is the value that distinguishes the "acceptance" region from the rejection region. The critical t -value, t c , is selected from a t -table depending on whether the test is one sided or two sided, on the level of significance 4 you specify and on the degrees of freedom, Once you have obtained a calculated t -value ( ˆ i t ) and a critical t -value ( t c ), the decision rule is as follows: 4 Also known as the probability of committing a Type I error . A Type I error consists in rejecting a true hypothesis, whereas a Type II error consists in accepting a false hypo thesis. The symbol α is also known as the size of the (statistical) test.

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ECON 301 - Introduction to Econometrics I April, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: Lecture Notes 10 If ˆ i t falls in the critical region (or rejection region), we reject the null hypothesis, that is, we say that the estimate of 1 is statistically significant. If ˆ i t falls in the acceptance region, that is to say if (taking the desired level of significance as 0.05) 0.025 0.025 ˆ i t t t (with T-k-1 degrees of freedom), we accept the null hypothesis, that is, we conclude with a probability of 95 percent that our estimate of 1 is not statistically significant. The acceptance and critical regions for 10 degrees of freedom and 0.05 level of significance are shown in Figure below ( ˆ 2.228 2.228 i t ).
ECON 301 - Introduction to Econometrics I April, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: Lecture Notes 11 t Test: Decision Rules for Two Sided Test Type of hypothesis H 0 : the null hypothesis H 1 : the alternative hypothesis Decision rule: reject H 0 if Two-tail * i i * i i /2, df t t Notes: * i is the hypothesized numerical value of i . | t | means the absolute value of t . /2 t means the critical t value at the α level of significance. df : degrees of freedom, (T − 2) for the two -variable model (including intercept), (T k 1) for the k independent variable model (including intercept). Example 1 Suppose that from a sample of size T =20, we estimate the following consumption function: ˆ 160.2 0.70 t t C Y (75.5) (0.21) The figures in parenthesis are the standard errors of the coefficients. For individual significance test of intercept term, we wish to test the hypothesis 0 0 : 0 H Against the alternative hypothesis 0 : 0 A H The t statistic for slope coefficient is as follows:

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ECON 301 - Introduction to Econometrics I April, 2012 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: Lecture Notes 12 0 0 ˆ 0 ˆ 160.2 2.12 75.5 ˆ ( ) t se Taking the level of significance as 0.05 ( =0.05), the critical values of t for T-k-1=20-1-1=18 degrees of freedom are: -t 0.025 =-2.101, and t 0.025 =2.101. Hence the acceptance region 0 0.025 0.025 ˆ t t t is 0 ˆ 2.101 2.101 t .
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