Chapter 8: Interval Estimates and Hypothesis Testing


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1 Chapter 8: Interval Estimates and Hypothesis Testing Chapter 8 Outline Clint s Assignment: Taking Stock Estimate Reliability: Interval Estimate Question o Normal Distribution versus the Student tdistribution: One Last Complication o Assessing the Reliability of a Coefficient Estimate: Applying the Student tdistribution Theory Assessment: Hypothesis Testing o Motivating Hypothesis Testing: The Cynic o Formalizing Hypothesis Testing: The Steps Summary: The Ordinary Least Squares (OLS) Estimation Procedure o Regression Model and the Role of the Error Term o Standard Ordinary Least Squares (OLS) Premises o Ordinary Least Squares (OLS) Estimation Procedure: Three Important Estimation Procedures o Properties of the Ordinary Least Squares (OLS) Estimation Procedure and the Standard Ordinary Least Square (OLS) Premises Each estimation procedure is unbiased. The estimation procedure for the coefficient value is the best linear unbiased estimation procedure (BLUE). Causation versus Correlation Chapter 8 Prep Questions 1. Run the following simulation and answer the questions posed. Summarize your answers by filling in the following blanks: [Link to MITLab 8P.1 goes here.] Actual Values From To Repetitions Between β x Var[e] Value Value From and to Values % % % 2. In the simulation you just ran (Question 1):
2 2 a. Using the appropriate equation, compute the variance of the coefficient estimate s probability distribution? b. What is the standard deviation of the coefficient estimate s probability distribution? c. Using the Normal distribution s rules of thumb, what is the probability that the coefficient estimate in one repetition would lie between: 1) 1.5 and 2.5? 2) 1.0 and 3.0? 3).5 and 3.5? d. Are your answers to part c consistent with your simulation results? 3. Recall the normal distribution. a. What is the definition of the normal distribution s z? b. Consider the regression results from Professor Lord s first quiz: Ordinary Least Squares (OLS) Dependent Variable: y Explanatory Variable(s): Estimate SE tstatistic Prob x Const Number of Observations 3 Sum Squared Residuals SE of Regression Recall the regression results from Professor Lord s first quiz: Ordinary Least Squares (OLS) Dependent Variable: y Explanatory Variable(s): Estimate SE tstatistic Prob x Const Number of Observations 3 Sum Squared Residuals SE of Regression a. Does the positive coefficient estimate suggest that studying more will improve a student s quiz score? Explain. Consider the views of a cynic: Cynic s view: Studying has no impact on a student s quiz score; the positive coefficient estimate obtained from the first quiz was just the luck of the draw. In fact, studying does not affect quiz scores. b. If the cynic were correct and studying has no impact on quiz scores, what would the actual coefficient, β x, equal?
3 3 c. Is it possible that the cynic is correct? To help you answer this question, run the following simulation: [Link to MITLab 8P.4 goes here.] Clint s Assignment: Taking Stock We shall begin by taking stock of where Clint stands. Recall the theory he must assess. Theory: Additional studying increases quiz scores. Clint s assignment is to assess the effect of studying on quiz scores: Project: Use data from Professor Lord s first quiz to assess the effect of studying on quiz scores. Clint uses a simple regression model to assess the theory. Quiz score is the dependent variable and number of minutes studied is the explanatory variable: y t = β Const + β x x t + e t where y t = Quiz score of student t x t = Minutes studied by student t e t = Error term for student t β Const and β x are the model s parameters. They incorporate the view that Professor Lord awards each student some points just for showing up; subsequently, the number of additional points each student earns depends on how much he/she studied: β Const represents the number of points Professor Lord gives a student just for showing up. β x represents the number of additional points earned for each additional minute of study. Since the values of β Const and β x are not observable, Clint adopted the econometrician s philosophy: Econometrician s Philosophy: If you lack the information to determine the value directly, estimate the value to the best of your ability using the information you do have. Clint used the results of the first quiz to estimate the values of β Const and β x by applying the ordinary least squares (OLS) estimation procedure to find the best fitting line: First Quiz Data T ( y y)( x x) Student x y b t t t= 1 x = = = = T ( xt x) t= 1 6 bconst = y bx x = 81 15= 81 18= 63 5
4 4 Clint s estimates suggest that Professor Lord gives each student 63 points for showing up; subsequently, each student earns 1.2 additional points for each additional minute studied. Clint realizes that he cannot expect the coefficient estimate to equal the actual value; in fact, he is all but certain that it will not. So now, Clint must address two related issues: Estimate Reliability: How reliable is the coefficient estimate, 1.2, calculated from the first quiz? That is, how confident should Clint be that the coefficient estimate, 1.2, will be close to the actual value? Theory Assessment: How confident should Clint be that the theory is correct, that studying improves quiz scores? We shall address both of these issues in this chapter. First, we consider estimate reliability. Estimate Reliability: Interval Estimate Question The interval estimate question quantifies the notion of reliability: Interval Estimate Question: What is the probability that the estimate, 1.20, lies within of the actual value? The general properties of the ordinary least squares (OLS) estimation procedure allow us to address this question. It is important to distinguish between the general properties and one specific application. Recall that the general properties refer to what we know about the estimation procedure before the quiz is given; the specific application refers to the numerical values of the estimates calculated from the results of the first quiz:
5 5 General Properties versus One Specific Application OLS Estimation Procedure: Estimate β Const and β x by Apply the estimation finding the b Const and b x that procedure once to the first minimize the sum of squared quiz s data: residuals Model: Before experiment y t = β Const + β x x t + e t After experiment Random Variable: Probability Distribution Mean[b x ] = β x Var[ e] Var[ b x ] = T ( x x) t= 1 t 2 b x OLS equations: = T t= 1 Const ( y y)( x x) T t t= 1 ( x x) b = y b x t x t 2 Estimate: Numerical Value b x = = = b Const = 81 15= 63 5 Mean and variance describe the center and spread of the estimate s probability distribution The estimates are random variables and a quiz can be viewed as an experiment. We cannot determine the numerical value of an estimate with certainty before the experiment (quiz) is conducted. What then do we know beforehand? We can describe the probability distribution of the estimate. We know that the mean of the coefficient estimate s probability distribution equals the actual value of the coefficient and its variance equals the variance of the error term s probability distribution divided by the sum of squared x deviations: Mean of Estimate s Probability Variance of Estimate s Distribution Equals Actual Value Probability Distribution Estimation Procedure Is Unbiased Determines the Reliability of the Estimate As Variance Decreases Reliability Increases
6 6 Both the mean and variance of the coefficient estimate s probability distribution play a crucial role: Since the mean of the coefficient estimate s probability distribution, Mean[b x ], equals the actual value of the coefficient, β x, the estimation procedure is unbiased; the estimation procedure does not systematically underestimate or overestimate the actual coefficient value. When the estimation procedure for the coefficient value is unbiased, the variance of the estimate s probability distribution, Var[b x ], determines the reliability of the estimate; as the variance decreases, the probability distribution becomes more tightly cropped around the actual value; consequently, it becomes more likely for the coefficient estimate to be close to the actual coefficient value. To assess his estimate s reliability, Clint must consider the variance of the coefficient estimate s probability distribution. But we learned that Clint can never determine the actual variance of the error term s probability distribution, Var[e]. Instead, Clint adopts a two step strategy for estimating the variance of the coefficient estimate s probability distribution: Step 1: Estimate the variance of the error term s Step 2: Apply the relationship between the probability distribution from the available variances of coefficient estimate s and information data from the first quiz error term s probability distributions EstVar[ e] = AdjVar[ Res's] Var[ e] Var[ b x ] = T SSR 54 2 = = = 54 ( xt x) Degrees of Freedom 1 t= 1 é ã EstVar[ e] 54 EstVar[ b x ] = = =.27 T ( x x) EstSD[ bx] = EstVar[ b x] =.27 =.5196 Unfortunately, there is one last complication before we can address the interval estimate question. t= 1 t
7 7 Normal Distribution versus the Student tdistribution: One Last Complication We begin by reviewing the normal distribution. Recall that the variable z played a critical role in using the normal distribution: Value of Random Variable Distribution Mean z = Distribution Standard Deviation = Number of Standard Deviations from the Mean In words, z equals the number of standard deviations the value lies from the mean. But Clint does not know what the variance and standard deviation of the coefficient estimate s probability distribution equal. That is why he must estimate them. Consequently, he cannot use the normal distribution to calculate probabilities. When the standard deviation is not known and must be estimated, the Student tdistribution must be used. The variable t is similar to the variable z; instead of equaling the number of standard deviations the value lies from the mean, t equals the number of estimated standard deviations the value lies from the mean: Value of Random Variable Distribution Mean t = Estimated Distribution Standard Deviation = Number of Estimated Standard Deviations from the Mean Recall that the estimated standard deviation is called the standard error; hence, Value of Random Variable Distribution Mean t = Standard Error = Number of Standard Errors from the Distribution Mean Probability Distribution of Random Variable Normal Student t Distribution Mean Figure 8.1: Normal and Student tdistributions Value of Random Variable
8 8 Like the normal distribution, the tdistribution is symmetric about its mean. Since estimating the standard deviation introduces an additional element of uncertainty, the Student tdistribution is more spread out than the normal distribution as illustrated in Figure 8.1. The Student tdistribution s spread depends on the degrees of freedom. As the number of degrees of freedom increase, we have more information; consequently, the tdistribution s spread decreases, moving it closer and closer to the normal distribution. Since the spread of the Student tdistribution depends on the degrees of freedom, the table describing the Student tdistribution is more cumbersome than the normal distribution table. Fortunately, our Econometrics Lab allows us to avoid the cumbersome Student tdistribution table. Assessing the Reliability of a Coefficient Estimate How reliable is the coefficient estimate, 1.2, calculated from the first quiz? That is, how confident should Clint be that the coefficient estimate, 1.2, will be close to the actual value? The interval estimate question to address this question: Interval Estimate Question: What is the probability that the coefficient estimate, 1.2, lies within of the actual coefficient value? We begin by filling in the first blank, choosing our close to value. The value we choose depends on how demanding we are; that is, our close to value depends on the range that we consider to be close to the actual value. For purposes of illustration, we shall choose 1.5; so we write 1.5 in the first blank. Interval Estimate Question: What is the probability that the coefficient estimate, 1.2, lies within 1.5 of the actual coefficient value? Figure 8.2 illustrates the probability distribution of the coefficient estimate and the probability that we wish to calculate. The estimation procedure we used to calculate the coefficient estimate, the ordinary least squares (OLS) estimation procedure is unbiased: Mean[b x ] = β x Consequently, we place the actual coefficient value, β x, at the center of the probability distribution.
9 9 Probability that the estimate is within 1.5 of the actual value Student tdistribution Mean[b x ] = β x b x β x 1.5 Actual Value = β x β x +1.5 Figure 8.2: Probability Distribution of Coefficient Estimate Close To Value Equals 1.5 As discussed above, we must use the Student tdistribution rather than the normal distribution since we must estimate the standard deviation of the probability distribution. The regression results from Professor Lord s first quiz provide the estimate: Ordinary Least Squares (OLS) Dependent Variable: y Explanatory Variable(s): Estimate SE tstatistic Prob x Const Number of Observations 3 Sum Squared Residuals SE of Regression Estimated Equation: Esty = x Interpretation of Estimates: b Const = 63: Students receive 63 points for showing up. b x = 1.2: Students receive 1.2 additional points for each additional minute studied. Critical Result: The coefficient estimate equals 1.2. The positive sign of the coefficient estimate, suggests that additional studying increases quiz scores. This evidence lends support to our theory. Table 8.1: Quiz Scores Regression Results
10 10 The standard error equals the estimated standard deviation. t equals the number of standard errors (estimated standard deviations) that the value lies from the distribution mean: t Value of Random Variable Distribution Mean = Standard Error = Number of Standard Errors from the Distribution Mean Since the distribution mean equals the actual value, we can translate 1.5 below and above the actual value into t s. Since the standard error equals.5196, 1.5 below and above the actual value translates into 2.89 standard errors below and above the actual value: 1.5 below actual value 1.5 above actual value = 2.89 SE's below actual value = 2.89 SE's above actual value To summarize, The probability that the estimate lies within 1.5 of the actual value. = The probability that the estimate lies within 2.89 SE s of the actual value. That is, between t s of 2.89 and 2.89 Figure 8.3 adds this information to the probability distribution graph. Probability that the estimate is within 1.5 of the actual value Student tdistribution Mean[b x ] = β x SE[b x ] = SE s 2.89 SE s β x 1.5 β x +1.5 Actual Value = β x t = 2.89 t = 2.89 Figure 8.3: Probability Distribution of Coefficient Estimate Close To Value Equals 1.5 b x
11 11 Econometrics Lab 8.1: Calculate Prob[Results IF H 0 True]. We can now use the Econometrics Lab to calculate the probability that the estimate is within 1.5 of the actual value by computing probabilities the left and right tails probabilities. 1 Left Tail: [Link to MITLab 8.1a goes here.] o The following information has been entered: Degrees of freedom: 1 t: 2.89 o Click Calculate. The left tail probability is approximately.11. Right Tail: [Link to MITLab 8.1b goes here.] o The following information has been entered: Degrees of freedom: 1 t: 2.89 o Click Calculate. The right tail probability is approximately.11. Since the Student tdistribution is symmetric, both the left and right tail probabilities equal.11. Hence, the probability that the estimate is within 1.5 of the actual value equals.78: 1.00 ( ) = Probability that the estimate is within 1.5 of the actual value Student tdistribution Mean[b x ] = β x SE[b x ] = SE s 2.89 SE s β x 1.5 β x +1.5 Actual Value = β t = 2.89 x t = 2.89 Figure 8.4: Probability Distribution of Coefficient Estimate Applying Student t Distribution b x
12 12 We can now fill in the second blank in the interval estimate question: Interval Estimate Question: What is the probability that the coefficient estimate, 1.2, lies within 1.5 of the actual coefficient value?.78 We shall turn our attention to assessing the theory. Theory Assessment: Hypothesis Testing Hypothesis testing allows Clint to assess how much confidence he should have in the theory. We begin by motivating hypothesis testing using the same approach as we took with Clint s opinion poll. We shall play the role of the cynic. Then, we shall formalize the process. Motivating Hypothesis Testing: The Cynic Recall that the theory suggests that a student s score on the quiz depends on the number of minutes he/she studies: Theory: Additional studying increases scores. Review the regression model: y t = β Const + β x x t + e t The theory suggests that β x is positive. Review the regression results for the first quiz: Ordinary Least Squares (OLS) Dependent Variable: y Explanatory Variable(s): Estimate SE tstatistic Prob x Const Number of Observations 3 Sum Squared Residuals SE of Regression Estimated Equation: Esty = x Interpretation of Estimates: b Const = 63: Students receive 63 points for showing up. b x = 1.2: Students receive 1.2 additional points for each additional minute studied. Critical Result: The coefficient estimate equals 1.2. The positive sign of the coefficient estimate, suggests that additional studying increases quiz scores. This evidence lends support to our theory. Table 8.2: Quiz Scores Regression Results
13 13 The estimate for β x, 1.2, is positive. We estimate that an additional minute of studying increases a student s quiz score by 1.2 points. This lends support to Clint s theory. But, how much confidence should Clint have in the theory? Does this provide definitive evidence that Clint s theory is correct or should we be skeptical? If β x = 0 Prob[b x > 0].50 0 Figure 8.5: Probability Distribution of Coefficient Estimate Could the Cynic Be Correct? To answer this question, recall our earlier hypothesis testing discussion and play the cynic. What would a cynic s view of our theory and the regression results be? Cynic s view: Studying has no impact on a student s quiz score; the positive coefficient estimate obtained from the first quiz was just the luck of the draw. In fact, studying has no effect on quiz scores; the actual coefficient, β x, equals 0. Is it possible that our cynic is correct? b x
14 14 Econometrics Lab 8.1: Assessing the Cynic s View We shall use our Could the Cynic Be Correct? simulation to show that it is. A positive coefficient estimate can arise in one repetition of the experiment even when the actual coefficient is 0. [Link to MITLab 8.2 goes here.] In the simulation, the default actual coefficient value is 0. Check the FromTo checkbox. Also, 0 is specified in the From list. In the To list, no value is specified; consequently, there is no upper FromTo bound. The FromTo Percent box will report the percent of repetitions in which the coefficient estimate equals 0 or more. Be certain that the Pause checkbox is cleared. Click Start and then after many, many repetitions, click Stop. In about half the repetitions, the coefficient estimate is positive; that is, when the actual coefficient, β x, equals 0, the estimate is positive about half the time. The histogram illustrates this. Now, we can apply the relative frequency interpretation of probability. If the actual coefficient were 0, the probability of obtaining a positive coefficient from one quiz would be about onehalf. Consequently, we cannot dismiss the cynic s view as absurd. To assess the cynic s view, we pose the following question: Question for the Cynic: What is the probability that the result would be like the one obtained (or even stronger), if studying actually has no impact on quiz scores? That is, what is the probability that the coefficient estimate from the first quiz would be 1.2 or more, if studying had no impact on quiz scores (if the actual coefficient, β x, equals 0)? Answer: Prob[Results IF Cynic Correct].
15 15 The magnitude of the probability determines the likelihood that the cynic is correct, the likelihood that studying has no impact on quiz scores: Prob[Results IF Cynic Correct] small Prob[Results IF Cynic Correct] large Unlikely that the cynic is correct Likely that the cynic is correct Unlikely that the studying has no impact Likely that the studying has no impact To compute this probability let us review what we know about the probability distribution of the coefficient estimate: OLS estimation If H 0 Standard Number of Number of procedure unbiased true error observations parameters é ã é ã Mean[ b x ] = β x = 0 SE[b x ] =.5196 DF = 3 2 = 1 Question for the Cynic: What is the probability that the coefficient estimate from the first quiz would be 1.2 or more, if studying had no impact on quiz scores (if the actual coefficient, β x, equaled 0)? Student tdistribution Mean = 0 SE =.5196 DF = Figure 8.6: Probability Distribution of Coefficient Estimate Prob[Results IF Cynic Correct] How can we answer this question? We turn to the Econometrics Lab. b x
16 16 Econometrics Lab 8.2: Using the Econometrics Lab to Calculate Prob[Results IF Cynic Correct] [Link to MITLab 8.3 goes here.] The appropriate information has been entered: Mean: 0 Value: 1.2 Standard Error:.5196 Degrees of Freedom: 1 Click Calculate. The probability that the estimate lies in the right tail equals.13. The answer to the question for the cynic is.13: Answer: Prob[Results IF Cynic Correct] =.13 In fact, there is an even easier way to compute the probability. We do not even need to use the Econometrics Lab to because the statistical software calculates this probability automatically. To illustrate this, we shall first calculate the tstatistic based on the premise that the cynic is correct, based on the premise that the actual value of the coefficient equals 0: Value of Random Variable Distribution Mean t = = = Standard Error.5196 = Number of Standard Errors from the Distribution Mean 1.2 lies standard errors from 0. Next, return to the regression results and focus attention on the row corresponding to the coefficient and on the tstatistic and Prob column. Ordinary Least Squares (OLS) Dependent Variable: y Explanatory Variable(s): Estimate SE tstatistic Prob x Const Number of Observations 3 Sum Squared Residuals SE of Regression Estimated Equation: Esty = x Interpretation of Estimates: b Const = 63: Students receive 63 points for showing up. b x = 1.2: Students receive 1.2 additional points for each additional minute studied. Critical Result: The coefficient estimate equals 1.2. The positive sign of the coefficient estimate, suggests that additional studying increases quiz scores. This evidence lends support to our theory. Table 8.3: Quiz Scores Regression Results
17 17 Two interesting observations emerge: First, the tstatistic column equals 2.309, the value of the tstatistic we just calculated; the tstatistic based on the premise that the cynic is correct and the actual coefficient equals 0. The tstatistic column reports the number of standard errors the coefficient estimate based on the premise that the actual coefficient equals 0. Second, the Prob column equals This is just twice the probability we just calculated using the Econometrics Lab: 2 Prob[Results IF Cynic Correct] = Prob Column 2.13 =.26 Student tdistribution Mean = 0 SE =.5196 DF = /2.2601/ Figure 8.7: Probability Distribution of Coefficient Estimate Tails Probability The Prob column is based on the premise that the actual coefficient equals 0 and then focuses on the two tails of the probability distribution where each tail begins 1.2 (the numerical value of the coefficient estimate) from 0. As Figure 8.7 illustrates, the value in the Prob column equals the probability of lying in the tails; the probability that the estimate resulting from one week s quiz lies at least 1.2 from 0 assuming that the actual coefficient, β x, equals 0. That is, the Prob column reports the tails probability: Tails Probability: The probability that the coefficient estimate, b x, resulting from one regression would lie at least 1.2 from 0 based on the premise that the actual coefficient, β x, equals 0. Consequently, we do not need to use the Econometrics Lab to answer the question that we pose for the cynic: Question for the Cynic: What is the probability that the coefficient estimate from the first quiz is 1.2 or more, if studying had no impact on quiz scores (if the actual coefficient, β x, equals 0)? Answer: Prob[Results IF Cynic Correct] b x
18 18 Student tdistribution Mean = 0 SE =.5196 DF = / Figure 8.8: Probability Distribution of Coefficient Estimate Prob[Results IF Cynic Correct] We can use the regression results to answer this question. From the Prob column we know that the tails probability equals We are only interested in the right tail, however, the probability that the coefficient estimate will equal 1.2 or more, if the actual coefficient equals 0. Since the Student tdistribution is symmetric, the probability of lying in one of the tails is The answer to the 2 question we posed to assess the cynic s view is.13: Tails Probability.2601 Prob[Results IF Cynic Correct] = = b x
19 19 Formalizing Hypothesis Testing: The Steps We formalized hypothesis testing in Chapter 4 when we considered Clint s public opinion poll. We shall follow the same steps here, with one exception. We add a Step 0 to construct an appropriate model to assess the theory. Theory: Additional studying increases quiz scores. Step 0: Formulate a model reflecting the theory to be tested. We have already constructed this model: y t = β Const + β x x t + e t y t = Quiz score β Const reflects points for showing up x t = Minutes studied β x reflects points for each minute studied e t = Error term The theory suggests that β x is positive. Step 1: Collect data, run the regression, and interpret the estimates. First Quiz Data Student x y b Const = Estimated points for showing up = 63 b x = Estimated points for each minute studied = 1.2 Ordinary Least Squares (OLS) Dependent Variable: y Explanatory Variable(s): Estimate SE tstatistic Prob x Const Number of Observations 3 Sum Squared Residuals SE of Regression Estimated Equation: Esty = x Interpretation of Estimates: b Const = 63: Students receive 63 points for showing up. b x = 1.2: Students receive 1.2 additional points for each additional minute studied. Critical Result: The coefficient estimate equals 1.2. The positive sign of the coefficient estimate, suggests that additional studying increases quiz scores. This evidence lends support to our theory. Table 8.4: Quiz Scores Regression Results
20 20 Step 2: Play the cynic and challenge the results; construct the null and alternative hypotheses. Cynic s view: Despite the results, studying has no impact on quiz scores. The results were just the luck of the draw. Now, we construct the null and alternative hypotheses. Like the cynic, the null hypothesis challenges the evidence; the alternative hypothesis is consistent with the evidence: H 0 : β x = 0 Cynic is correct: Studying has no impact on a student s quiz score. H 1 : β x > 0 Cynic is incorrect: Additional studying increases quiz scores. Step 3: Formulate the question to assess the cynic s view and the null hypothesis. Question for the Cynic: Generic Question: What is the probability that the results would be like those we actually obtained (or even stronger), if the cynic is correct and studying actually has no impact? Specific Question: The regression s coefficient estimate was 1.2: What is the probability that the coefficient estimate in one regression would be 1.2 or more if H 0 were actually true (if the actual coefficient, β x, equals 0)? Answer: Prob[Results IF Cynic Correct] or Prob[Results IF H 0 True] The magnitude of this probability determines whether we reject the null hypothesis: Prob[Results IF H 0 True] small Prob[Results IF H 0 True] large Unlikely that H 0 is true Likely that H 0 is true Reject H 0 Do not reject H 0
21 21 Step 4: Use the general properties of the estimation procedure, the probability distribution of the estimate, to calculate Prob[Results IF H 0 True]. OLS estimation If H 0 Standard Number of Number of procedure unbiased true error observations parameters é ã é ã Mean[b x ] = β x = 0 SE[b x ] =.5196 DF = 3 2 = 1 We have already calculated this probability. First, we did so using the Econometrics Lab. Then, we noted that the statistical software had done so automatically. We need only divide the tails probability, as reported in the Prob column of the regression results, by 2:.2601 Prob[Results IF H 0 True] =.13 2 The probability that the coefficient estimate in one regression would be 1.2 or more if H 0 were actually true (if the actual coefficient, β x, equals 0) is.13. Step 5: Decide on the standard of proof, a significance level. The significance level is the dividing line between the probability being small and the probability being large. Prob[Results IF H 0 True] Prob[Results IF H 0 True] less than significance level greater than significance level Prob[Results IF H 0 True] small Prob[Results IF H 0 True] large Unlikely that H 0 is true Likely that H 0 is true Reject H 0 Do not reject H 0 Recall that the traditional significant levels used in academia are 1, 5, and 10 percent. Obviously,.13 is greater than.10. Consequently, Clint would not reject the null hypothesis that studying has no impact on quiz scores even with a 10 percent significance level.
22 22 Summary: The Ordinary Least Squares (OLS) Estimation Procedure Regression Model and the Role of the Error Term Now, let us sum up what we have learned about the ordinary least squares (OLS) estimation procedure: y t = β Const + β x x t + e t y t = Dependent variable x t = Explanatory variable e t = Error term t = 1, 2,, T T = Sample size The error term is a random variable; it represents random influences. The mean of the each error term s probability distribution equals 0: Mean[e t ] = 0 For each t = 1, 2,, T Standard Ordinary Least Squares (OLS) Premises Error Term Equal Variance Premise: The variance of the error term s probability distribution for each observation is the same; all the variances equal Var[e]: Var[e 1 ] = Var[e 2 ] = = Var[e T ] = Var[e] Error Term/Error Term Independence Premise: The error terms are independent: Cov[e i, e j ] = 0. Knowing the value of the error term from one observation does not help us predict the value of the error term for any other observation. Explanatory Variable/Error Term Independence Premise: The explanatory variables, the x t s, and the error terms, the e t s, are not correlated. Knowing the value of an observation s explanatory variable does not help us predict the value of that observation s error term.
23 23 Ordinary Least Squares (OLS) Estimation Procedure: Three Important Estimation Procedure There are three important estimation procedures embedded within the ordinary least squares (OLS) estimation procedures. A procedure to estimate the Values of the regression parameters, β x and β Const : T ( y y)( x x) b b y b x t t t= 1 x = and Const = T x 2 ( xt x) t= 1 Variance of the error term s probability distribution, Var[e]: SSR EstVar[ e ] = Degrees of Freedom Variance of the coefficient estimate s probability distribution, Var[b x ]: EstVar[ e] EstVar[ b x ] = T 2 ( x x) t= 1 t Properties of the Ordinary Least Squares (OLS) Estimation Procedure and the Standard Ordinary Least Square (OLS) Premises When the standard ordinary least square (OLS) premises are met: Each estimation procedure is unbiased; each estimation procedure does not systematically underestimate or overestimate the actual value. The ordinary least squares (OLS) estimation procedure for the coefficient value is the best linear unbiased estimation procedure (BLUE).
24 Causation versus Correlation Our theory and Step 0 illustrate the important distinction between causation and correlation: Theory: Additional studying increases quiz scores. Step 0: Formulate a model reflecting the theory to be tested. y t = β Const + β x x t + e t y t = Quiz score β Const reflects points for showing up x t = Minutes studied β x reflects points for each minute studied e t = Error term The theory suggests that β x is positive. Our model is a causal model. An increase in studying causes a student s quiz score to increase: Increase in studying (x t ) Causes Quiz score to increase (y t ) Correlation results whenever a causal relationship describes the reality accurately. That is, when additional studying indeed increases quiz scores, studying and quiz scores will be (positively) correlated: Knowing the number of minutes a student studies allows us to predict his/her quiz score. Knowing a student s quiz score helps us predict the number of minutes he/she has studied. More generally, a causal model that describes reality accurately implies correlation: Causation Implies Correlation Beware that correlation need not imply causation, however. For example, consider precipitation in the Twin Cities, precipitation in Minneapolis and precipitation in St Paul. Since the cities are near each other precipitation in the two cities are highly correlated. When it rains in Minneapolis it also always rains in St Paul also and vice versa. But there is no causation involved here. Rain in Minneapolis does not cause rain in St. Paul nor does rain in St. Paul cause rain in Minneapolis. The rain is caused by the weather system moving over the cities. In general, the correlation of two variables need not imply that a causal relationship exists between the variables: Correlation Need Not Imply Causation 24
25 25 Appendix 8.1: Student tdistribution Table Right Tail Critical Values α: Right Tail Probability 0 Figure 8.9: Student tdistribution Right Tail Probabilities Degrees of Freedom α = 0.10 α = 0.05 α = α = 0.01 α = t
26 Table 8.5: Right Tail Critical Values for the Student tdistribution 26
27 27 Appendix 8.2 Assessing the Reliability of a Coefficient Estimate Using the Student tdistribution Table We begin by describing the Student tdistribution table; a portion of it appears in Table 8.6: Degrees of Freedom α = 0.10 α = 0.05 α = α = 0.01 α = Table 8.6: Right Tail Critical Values for the Student tdistribution The first column represents the degrees of freedom. The numbers in the body of the table are called the critical values. A critical value equals the number of standard errors a value lies from the mean. The top row specifies α s value of, the right tail probability. Since the tdistribution is symmetric, the left tail probability also equals α. The probability of lying within the tails, in the center of the distribution, is 1 2α. This no doubt sounds confusing, but everything should become clear after we show how Clint can use this table to answer the interval estimate question. 1 2α Student tdistribution α α Critical Value SE Critical Value SE Distribution Mean Estimate Figure 8.10: Student tdistribution Illustrating the Probabilities Interval Estimate Question: What is the probability that the estimate, 1.2, lies within of the actual value? Let us review the regression results from Professor Lord s first quiz: Coefficient Estimate = b x = 1.2 Standard Error of Coefficient Estimate = SE[b x ] =.5196
28 28 Next, we shall modify Figure 8.10 to reflect our specific example. Focus on Figure We are interested in the coefficient estimate; consequently, we replace the horizontal axis label by substituting b x for Estimate. Also, we know that the estimation procedure Clint uses, the ordinary least squares (OLS) estimation procedure, is unbiased; hence, the distribution mean equals the actual value. We can replace the Distribution Mean with the actual coefficient value, β x. 1 2α Student tdistribution α α Critical Value SE Critical Value SE b x β x Figure 8.11: Student tdistribution Illustrating the Probabilities for Coefficient Estimate Now, let us help Clint fill in the blanks. When using the table we begin by filling in the second blank rather than the first. Second Blank: Choose α to specify the tail probability. Clint must choose a value for α. As we shall see, the value he chooses depends on how demanding he is. For example, suppose that Clint believes that a.80 probability of the estimate lying in the center of the distribution, close to the mean, is good enough. He would then choose an α equal to.10. To understand why, note that when α equals.10, the probability of the estimate lying in the right tail would be.10. Since the tdistribution is symmetric, the probability of the estimate lying in the left tail would be.10 also. Therefore, the probability that the estimate lies in the center of the distribution would be.80; accordingly, we write.80 in the second blank. What is the probability that the estimate, 1.2, lies within of the actual value?.80 First Blank: Calculate tail boundaries.
29 29 The first blank quantifies what close to means. The standard error and the Student tdistribution table allow us to fill in the first blank. To do so, we begin by calculating the degrees of freedom. Recall that the degrees of freedom equal 1: Degrees of Freedom = Sample Size = 3 2 = 1 Number of Estimated Parameters Degrees of Freedom α = 0.10 α = 0.05 α = α = 0.01 α = Table 8.7: Right Tail Critical Values for the Student tdistribution α Equals 0.10 and Degrees of Freedom Equals 1 Clint chose a value of α equal to.10. The table indicates that the critical value for α =.10 with 1 degree of freedom is The probability that the estimate falls within standard errors of the mean is.80. Next, the regression results report that the standard error equals.5196: SE[b x ] =.5196 After multiplying the critical value given in the table, 3.078, by the standard error,.5196, we can fill in the first blank: = Student tdistribution Critical Value SE Critical Value SE b x = = 1.6 β x 1.6 β x β x +1.6 Figure 8.12: Student tdistribution Calculations for an α Equal to.10
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