Economics 306 Midterm 1 Spring 2015 100 Points 100 Minutes Instructions Answer the following questions as clearly as you can Please circle your answers Partial credit is available but we cannot give partial credit unless you show your work You do not need to write out every detail but give a clear indication of how you are making your calculations For problems with many parts where you use information from earlier in the problem to answer the question you can receive full credit if you do the correct work but use the wrong numbers The clearer you are in your explanation the more straightforward this part of the grading will be You may use any type of hand held calculator All other electronic devices are prohibited DO NOT OPEN UNTIL I START THE TEST Part 1 Each question in this part is worth 5 points 70 points total You have collected data on two variables The x variable is the number of hours spent studying for an exam and the y variable is the score on the exam You have 24 observations You calculate the following values using Excel in order to calculate the OLS regression by hand Also in case this didn t make it onto your formula sheet it was on a practice problem note that it can be shown that the Sum of Squares Estimated 1 What is 2 What is the sample variance of Y 3 What is the sample standard deviation of thus 4 What is the sample covariance of X and Y 5 What is the value of 6 Interpret using words the meaning of in this particular regression As the numbers of hours of study increases by 1 the exam score is predicted to increase by 7941 points 7 What is the value of 8 What is the value of This equals zero due to the definition of Xbar 9 Suppose one observation call it observation 0 in the sample is X0 10 Y0 70 What is the value of The error is the difference between the actual and predicted values 70 66 40 3 60 10 What is the value of That is the variance of the regression SUBSTANTIAL ROUNDING ERRORS ARE GOING TO START TO CREEP IN NOW STUDENTS ARE NEVER TO BE PENALIZED FOR DOING THE RIGHT WORK BUT GETTING NUMBERS A LITTLE DIFFERENT FROM WHAT I HAVE The clue to use is critical now SSR SST SSE 45000 21441 18 23558 82 Finally 11 What is the variance of 12 What is the test statistic for a null hypothesis that 50 13 Will you reject the above null hypothesis at a level of 05 Why or why not No The critical value is larger than this test statistic in absolute value 2 07 is the critical value to be precise 14 What percentage of the variation of Y is explained by X Part 2 30 points total 1 10 points Claim All else equal as the variance of the independent variable increases the OLS estimator becomes more precise a True or False 2 points True b Use math to explain your answer for a 4 points One version of the formula for the variance of the estimator betahat is As the variance of X rises the denominator rises making the variance of betahat smaller or equivalently the estimator becomes more precise c Use English to explain the intuition for your answer for a 4 points We are seeking to explain the dependent variable Y with X The wider the range of values we have in our dataset for X and the variance is the measure of this variation of X the more information we will have about how X and Y are connected which improves our precision As a useful analogy it s like trying to balance a beam of wood Y on some support X If the support is very narrow like a pencil tip the beam of wood will be very unstable and the beam will tilt one way of another However if the support is broad like a table top the beam will be stable and not likely to tip Same with 2 12 points total 4 points each part Compare the following two OLS regressions i ii The sign distinguishes the fact that the values you d calculate for equation ii may differ from i The values of the Y variable have been multiplied by and the value of the X variables have been multiplied by 2 How do the values of the following terms derived from equation ii differ from the equivalent terms for equation 1 You need to give us some indication of how you arrive at your answers I recommend answering these in the order given If a person correctly identifies the direction but was unable to quantify the amount of change that is worth 1 5 points each A decent but failed attempt to mathematically determine the change is worth 2 points assuming they are going in the right direction If they do the math right but don t say at the end slope decreases etc that should not be penalized a b Slope decreases by a factor of 8 Intercept decreases by factor of 4 c SSR SSR decreases by a factor of 16 3 8 points State three reasons why one would not expect all the stochastic error terms i in a population regression model to equal zero I do not mean the average error term which of course will equal zero These may include the following A student does not need to get everything to get full marks Incomplete Theory Simplicity Ignorant of some factors that determine y Unpredictability of human behavior idisyncracies Misspecification of the functional form of relationship of x and y We write models that focus on core variables and ignore peripheral ones Occam s razor All else equal we prefer models that are simple but that simplicity may cost accuracy Whether that s a good tradeoff depends on the importance of that tradeoff Incomplete Information predicting y We may not have all the variables we need to describe the model If we use a proxy variable that proxy may not be as accurate as the missing variable in Cutoff Points for the t distribution df p 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 21 22 23 24 25 26 27 28 29 30 40 60 120 infinity 0 7 0 727 0 617 0 584 0 569 0 559 0 553 0 549 0 546 0 543 0 542 0 54 0 539 0 538 0 537 0 536 0 535 0 534 0 534 0 533 0 532 0 532 0 532 0 531 0 531 0 531 0 531 0 53 0 53 0 53 0 529 0 527 0 526 0 524 0 6 0 325 0 289 0 277 0 271 0 267 0 265 0 263 0 262 0 261 0 26 0 26 0 259 0 259 0 258 0 258 0 258 0 257 0 257 0 257 0 257 0 256 0 256 0 …