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Sided alternative way 2 take the following steps 1 2

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Unformatted text preview: g Ho when it is true): e.g. α = 5%. Then, you can proceed in one of two ways: 1 way 1 Melissa Tartari (Yale) Econometrics 24 / 41 2- Sided Alternative Consider the case in which Ho : βj = 0 is tested against H1 : βj 6= 0. Let tβ be the value of the statistics computed using sample ˆ j information. First, we choose a signi…cance level α (probability of rejecting Ho when it is true): e.g. α = 5%. Then, you can proceed in one of two ways: 1 2 way 1 way 2 Melissa Tartari (Yale) Econometrics 24 / 41 2- Sided Alternative, Way 1 Take the following steps: 1 for the chosen α, determine a critical value c where c is such that α α Pr ob (jT j > cα ) = α where T is a RV distributed as tn K 1 2 compare tβ to cα and reject Ho if tβ > cα (equivalently if ˆ ˆ j j tβ < cα or tβ > cα ) ˆ ˆ j j Two-sided alternative: Testing 1 Density of T Area= − cα if Melissa Tartari (Yale) t βˆ j 0 α 2 cα reject Ho Econometrics 25 / 41 2- Sided Alternative, Way 2 Take the following steps: 1 2 compute Pr ob jT j > tβ and call it p : this is known as the ˆ j p value since p is the smallest α at which Ho would be rejected, if your choice of α is bigger than p you reject Ho Two-sided alternative: Testing 2 Density of T Area= t βˆ j if Melissa Tartari (Yale) α>p 0 p 2 t βˆ j reject Ho Econometrics 26 / 41 Summary for Simple Hypothesis Given Ho and H1 , we choose a signi…cance level α which then determines a critical value cα . Melissa Tartari (Yale) Econometrics 27 / 41 Summary for Simple Hypothesis Given Ho and H1 , we choose a signi…cance level α which then determines a critical value cα . We then compare the value of the t statistic with cα and Ho is either rejected or not rejected at the given signi…cance level α depending on the results of this comparison. Melissa Tartari (Yale) Econometrics 27 / 41 Summary for Simple Hypothesis Given Ho and H1 , we choose a signi…cance level α which then determines a critical value cα . We then compare the value of the t statistic with cα and Ho is either rejected or not rejected at the given signi…cance level α depending on the results of this comparison. A summary of the rejection rules is given in the table below (col 2). Melissa Tartari (Yale) Econometrics 27 / 41 Summary for Simple Hypothesis Given Ho and H1 , we choose a signi…cance level α which then determines a critical value cα . We then compare the value of the t statistic with cα and Ho is either rejected or not rejected at the given signi…cance level α depending on the results of this comparison. A summary of the rejection rules is given in the table below (col 2). The rejection rule is determined by our choice H1 while the rejection level cα is determined by our choice of α. Melissa Tartari (Yale) Econometrics 27 / 41 Summary for Simple Hypothesis Rather than testing at di¤erent α, it is more informative to answer the following question: given the observed value of the t statistic, what is the smallest α at which Ho would be rejected? Melissa Tartari (Yale) Econometrics 28 / 41 Summary for Simple Hypothesis Rather than testing at di¤erent α, it is more informative to answer the following question: given the observed value of the t statistic, what is the smallest α at which Ho would be rejected? This level is know as the p value: it summarizes the strength of the empirical evidence against Ho . In order to compute p values we use the STATA command ttail or look directly at the output of the command regress which reports the p value for testing Ho : βj = 0 against a two-sided alternative. (col 3&4). H1 Rejection Rule p βj > 0 t β > cα ˆ p /2 p /2 < α βj < 0 t β < cα ˆ p /2 p /2 < α p p<α βj 6 = 0 Melissa Tartari (Yale) j j t β > cα ˆ j Econometrics value Rejection Rule 28 / 41 Summary for Simple Hypothesis Interpret the p value as the probability of observing a t statistics as extreme as we did if Ho is true ) small p values are evidence against Ho while large p values provide little evidence against Ho . Melissa Tartari (Yale) Econometrics 29 / 41 Summary for Simple Hypothesis Interpret the p value as the probability of observing a t statistics as extreme as we did if Ho is true ) small p values are evidence against Ho while large p values provide little evidence against Ho . EXAMPLE: if p value = 0.10, then we would observe a value of the t statistics as extreme as we did in 10% of all random samples when Ho is true; this is pretty weak evidence against Ho ) we fail to reject Ho at any signi…cance level α < 0.10 Melissa Tartari (Yale) Econometrics 29 / 41 Summary for Simple Hypothesis Once the p value has been computed, a classical test can be carried out at any α; computing p values for one-sided H1 is also simple: just divide the two-sided p value by 2. Melissa Tartari (Yale) Econometrics 30 / 41 Summary for Simple Hypothesis Once the p value has been computed, a classical test can be carried out at any α; computing p values for one-sided H1 is also simple: just divide the two-sided p value by 2. EXAMPLE: Let tβ be the computed value of our statistics. STATA ˆ j gives us p = P...
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