Chapter 3 practice MA
Consider the OLS estimator for the standard error of the slope coefficient. Which of the following statement(s) is (are) true? i: The standard error will be positively related to the residual variance ii: The standard error will be negatively related to the dispersion of the observations on the explanatory variable about their mean value iii: The standard error will be negatively related to the sample size iv: The standard error gives a measure of the precision of the coefficient estimate. a. ii and iv b. i and iii c. i, ii, iii d. all of the above
All of statements (i) to (iv) are true. The bigger the residual variance is, the bigger must be the RSS, and therefore the further away are the points from the line. Therefore, the bigger the residual variance is, the bigger will be the coefficient standard errors. This can bee seen since the term "s" appears positively in the standard error formulae for the intercept and the slope. The more dispersed are the observations on the explanatory variable (x) about its mean value, the more precisely the coefficient estimates can be calculated since we would have information about the relationship between y and x over a wider range of values for x. In the formulae, the variation of x about its mean value enters into the denominator for both the slope and the intercept standard errors, so the bigger the dispersion is, the smaller will be the standard errors. The bigger the sample size, the more pieces of information are available from which to estimate the model parameters. The number of observations appears explicitly in the formula for the intercept standard error and implicitly in the formula for the slope standard error. In the latter case, the standard error is inversely related to the sample size since the sum of the squares of the observations on x about their mean value appears in the denominator, and the larger the sample size is, the more terms will be included in this sum. d
If an estimator is said to have minimum variance, which of the following statements is NOT implied? a. The probability that the estimate is a long way away from its true value is minimised b. The estimator is efficient c. such an estimator would be termed "best" d. such an estimator will always be unbiased
An estimator that has minimum variance would also be defined as efficient and "best" - these terms are equivalent to one another. A minimum variance estimator means that the sampling variation in the parameter estimates between one sample and another will be minimised. This is also equivalent to stating that the probability that the estimate for any given sample is a long way off from its true value will be minimised. An estimator can have minimum variance but be a biased estimator. Typically there is an implicit trade off between choosing an unbiased but inefficient estimator and choosing an estimator with a smaller variance that is biased.
Which of the following statements is INCORRECT concerning the classical hypothesis testing framework? a. If the null hypothesis is rejected, the alternative is accepted b. The null hypothesis is the statement being tested while the alternative encompasses the remaining outcomes of interest c. The test of significance and confidence interval approaches will always give the same conclusions d. Hypothesis tests are used to make inferences about the population parameters.
Hypothesis tests are used to make statements about the plausibility of certain values for the population parameters given the estimates made from the sample. By definition, the null hypothesis is the statement being tested while the alternative encompasses other outcomes of interest. The test of significance and confidence interval approaches will always give the same answer (so long as a fixed significance level is used for both) since one can be viewed as just a rearrangement of the other. It is never said that the alternative hypothesis is accepted. The reason that this is not done is that, in general terms, it is possible to reject a null hypothesis without the alternative hypothesis being correct. Therefore a is the only incorrect statement. a.
Which of the following statements is TRUE concerning OLS estimation? a. OLS minimises the sum of the vertical distances from the points to the line b. OLS minimises the sum of the squares of the vertical distances from the points to the line c. OLS minimises the sum of the horizontal distances from the points to the line d. OLS minimises the sum of the squares of the horizontal distances from the points to the line
OLS minimises the sum of the squares of the vertical distances from the points to the line. The reason that vertical rather than horizontal distances are chosen is due to the set up of the classical linear regression model that assumes x is non-stochastic. Therefore, the question becomes one of how to find the best fitting values of y given the values of x. If we took horizontal distances, this would mean that we were choosing fitted values for x, which wouldn't make sense since x is fixed. The reason that squares of the vertical distances are taken rather than the vertical distances themselves is that some of the points will lie above the fitted line and some below, cancelling each other out. Therefore, a criterion that minimised the sum of the distances would not give unique parameter estimates since an infinite number of lines would satisfy this. b
Which of the following are alternative names for the dependent variable (usually denoted by y) in linear regression analysis? (i) The regressand (ii) The regressor (iii) The explained variable (iv) The explanatory variable a. ii and iv b. i and iii c. i ii, iii d. all of the above
The correct answer is b, since regressand and explained variable are alternative names for the variable whose movements we are trying to explain. The regressor or explanatory variable are names for x, the variable that is doing the explaining in the model.
The residual from a standard regression model is defined as a. The difference between the actual value, y, and the mean, y-bar b. The difference between the fitted value, y-hat, and the mean, y-bar c. The difference between the actual value, y, and the fitted value, y-hat d. The square of the difference between the fitted value, y-hat, and the mean, y-bar.
The residual is defined as the difference between the actual value y and the fitted value, y-hat. c
3. Which of the following statements is TRUE concerning the standard regression model? a. y has a probability distribution b. x has a probability distribution c. the disturbance is assume to be correlated with x d. for an adequate model, the reisdual (u-hat) will bezero for all sample data points.
a. Since y depends on u as well as x, and since u is a random variable, y will also be a random variable. x is assumed to be non-stochastic, i.e. to be fixed and it is therefore not a random variable. Since x is assumed to be non-stochastic, it cannot be correlated with a random variable u, otherwise it would be stochastic! A good model would be one where the residuals are as close to zero as possible. However, unless there is a perfect relationship between y and x (i.e. all of the points lie on a straight line), the residuals cannot all be zero.
Which of the following is an equivalent expression for saying that the explanatory variable is "non-stochastic"? a. The explanatory variable is partly random b. The explanatory variable is fixed in repeated samples c. The explanatory variable is correlated with the errors d. The explanatory variable always has a value of one
b. The explanatory variable is fixed in repeated samples The word "stochastic" means random, so non-stochastic means non-random! One of the classical linear regression model assumptions is that the explanatory variable x is non-stochastic or fixed in repeated samples. These are approximately equivalent expressions, although the latter is a slightly stronger statement. Note that "fixed in repeated samples" does not mean that its value is always the same (answer d), and also that this prevents x from being partly random (stochastic) or correlated with the errors (also implying that x is stochastic).
Which of the following are alternative names for the independent variable (usually denoted by x) in linear regression analysis? (i) The regressor (ii) The regressand (iii) The causal variable (iv) The effect variable a. ii and iv b. i and iii c. i ii and iii all of the above
b. The independent variable, usually denoted by x, is also known as the regressor or the causal variable. The regressand and effect variable are alternative names for y.
If an estimator is said to be consistent, it is implied that a. On average, the estimated coefficient values will equal the true values b. The OLS estimator is unbiased and no other unbiased estimator has a smaller variance c. The estimates will converge upon the true values as the sample size increases d. The coefficient estimates will be as close to their true values as possible for small and large samples.
c. The estimates will converge upon the true values as the sample size increases By definition, a consistent estimator is one where the sample estimates converge on their true (population) values as the sample size increases. Answer a is the definition for an unbiased estimator, while b is the result that is proved by the Gauss-Markov theorem. Answer d is a slightly different way of stating the unbiasedness property.
Which of the following statements concerning the regression population and sample is FALSE? a. the population is the total collection of all items of interest b. the population can be infinite c. the theory, the sample could be larger than the population d. a random sample is one where each individual item from the population is equally likely to be drawn
c. the theory, the sample could be larger than the population By definition, the population is indeed the collection of all items of interest, and this can be either infinite or finite depending on the context. Also by definition, a random sample is one where each item from the population is equally likely to be drawn. It is of course impossible for the sample to be larger than the population, since the sample takes just some items from the population.
Which of the following statements is true concerning the population regression function (PRF) and sample regression function (SRF)? a. the PRF is the estimated model b. the PRF is used to infer likely values of the SRF c. whether the model is good can be determined by comparing the SRF and the PRF d. the PRF is a description of the process though to be generating the data
d. The PRF is a description of the process thought to be generating the data. The PRF is the true population model for the relationship between the variables x and y. Some researchers draw a distinction between the PRF and data generating process, but the two terms have been used synonymously on this course. The sample is used to estimate a SRF, which is used to determine what are the likely values of the population parameters described by the PRF. Therefore a, b, and c are false and d is a true statement.
