Kinh tế học - Chapter III: Statistic inference and hypothesis testing

Chapter IIIStatistic Inference and Hypothesis Testing APPLIED ECONOMETRICS COURSENGUYEN BA TRUNG - 2016I. THE DISTRIBUTION OF THE PARAMETERSAssumpt 6. Ui ~ N(0, 2)Theorem 4.1: Normal Distribution of the parametors I. THE DISTRIBUTION OF THE PARAMETERS Under the assumptions from 1-6, we have the following theorem:Theorem 4.2: Student DistributionII. HYPOTHESES TESTING (1). Manifest the hypothesis:(2). Compute t-statistics:(3). Search to find t (n-k-1), for example, α=5%, n-k-1=28, t0.05 (28)=1.701 ( excel: TINV(0.1,28)) (4). Decision Rule: Nếu t > t (n-k-1) thì có thể bác bỏ giả thiết H0Nếu t t0.05 (8)=1.86, we therefore can reject the null hypothesis H0. In other word, we can accept the alternative hypothesis H1: 1 > 0II. HYPOTHESES TESTING 2.2. Testing against One-Sided Alternatives: Left Hand Side(1). Suppose the hypothesis:(2).Compute t-statistics:(3).Search to find t (n-k-1)(4). Decision Rule : Nếu t t/2(n-k-1), we can reject the null hypothesis H0if  t  t/2(n-k-1), we can accept the null hypothesis H0II. HYPOTHESES TESTING Example: Whether income affects to expenditure?Manifest the hypothesis: Compute s-statistics:With  = 5%, we have: t0.025(n-k-1) = t0.025(8) = 2.306Because  t =14.243 > t0.025(8) = 2.36, we can reject the null hypothesis H0: β1= 0. This means that income significantly impacts to expenditure.Solution:Example: expenditure.wf2.4. Testing Other Hypotheses about parameterSuppose that there is a research showing that β1= 0.3, Whether this conclusion is true or not? Set up the null hypothesis: Compute s-statistics:Search to find t/2(n-k-1) Decision: If  t > t/2(n-k-1) => Reject H0 If  t  t/2(n-k-1) => Accept H0Solution: t-statistics: Search to find: t0.025(n-k-1)=t0.025(8) = 2.306 Because  t =5.85 > t/2(8) = 2.36, we can reject H0: β1= 0.3 with  = 5%. (1). Null hypothesis:2.5. Compute P-value for t-tests With confident level α, P-value is the probability of mistake if we reject the null hypothesis H0 P-value: Prob(|T|>|t|) Thus, if small P-value provides the evidence against H0, and if large P-value provide the evidence against H1 Normally, we often choose confident level at 1%, 5%, and 10%. Ví dụ: expenditure.wfConfident Interval of the parameters 1; 2 Confident Interval of error term UiIII. Confident IntervalExample: expenditure (continue)Let’s compute the confident intervals for the parameters at confident level 95%? Because 1-  =0.95 ➨ = 0.05 ➨ /2 = 0.025 t 0.025(8) =2.306 [TINV(0.05,8)]hay (9.6643 F(k,n-k-1,α), we can reject H0. If F ≤ F(k,n-k-1,α), we can not reject H0.(3). Search to find: F(k,n-k-1,α) Example: continuted Compute F-statistics: With α = 5%, search to find Fα (k, n-k-1), we have: F0.05 (1,8)= 5.32 Because F = 202.86 > F0.05 (1,8)= 5.32, we can reject H0EVIEWs provide the p-value of F-test(next slide)Thí dụ: expenditure.wfV. Test the simple linear combination of the parameters(1). Assume that we want to test:(2). Compute t-statistics as: Where:(3). With α = 5%, search to find: tα/2(n-k-1)(4). Decision: If |t| > tα/2(n-k-1), We can reject H0Example:Twoyears.wfWe have the following model:Where: Jc: trình độ cao đẳng (2 năm đại học) Unive: trình độ đại học (4 năm đại học) Exper: kinh nghiệm (tháng)Question: whether wage differs between who those graduated from colleague and university? From the estimate result, the different wage: Is this difference statistically significant? (1). Set up the hypothesis: This implies: (2). Then we have:(3). With α = 5%, t0.025(6759) = 1.96(4). Because |t| = 1.42 , we can reject H0 Example: BWGHT.wf Compute F-statistics as: Search to find: F0.05(2,1185) = 3 Because F =1.42 < F0.05 (2,1185) = 3  reject H0. In other words, neither mother education and father education affect statistically significant to birth weight. Ví dụ:BWGHT.wf END OF THE CHAPTER III