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regression analysis

Applied Regression Analysis And Generalized Linear Models 3rd Edition By John Fox - Solutions

*For the statistic t ¼ Bj ( βj SE ffiffiffiffi vjj p to have a t-distribution, the estimators Bj and SE must be independent. [Here, vjj is the jth diagonal entry of ðX0 XÞ(1.] The coefficient Bj is the jth element ofb, and SE ¼
*Using Equation 9.12 (page 214), show that the maximized likelihood for the linear model can be written as L ¼ 2πe e0 en& '(n=2
Using Duncan’s regression of occupational prestige on income and education, and performing the necessary calculations, verify that the omnibus null hypothesis H0:β1 ¼ β2 ¼ 0 can be tested as a general linear hypothesis, using the hypothesis matrix L ¼ 010 001 $ %and right-hand-side vector
*Consider the model Yi ¼ β0 þ β1xi1 þ β2xi2 þ εi. Show that the matrix V(1 11(see Equation 9.16 on page 218) for the slope coefficients β1 and β2 contains mean deviation sums of squares and products for the explanatory variables; that is, V(1 11 ¼Px*2 i1 Px*i1x*P i2 x*i1x*i2 Px*2 i2$
*Show that Equation 9.20 (page 222) for the confidence interval for β1 can be written in the more conventional form B1 ( ta; n(3 SE ffiffiffiffiffiffiffiffiffiffiffiffiffiffi Px*2 i1 1 ( r2 12 s £ β1 £ B1 þ ta; n(3 SE ffiffiffiffiffiffiffiffiffiffiffiffiffiffi Px*2 i1 1 ( r2 12 s
Using Figure 9.2 (on page 223), show how the confidence interval–generating ellipse can be used to derive a confidence interval for the difference of the parameters β1 ( β2.Compare the confidence interval for this linear combination with that for β1 þ β2. Which combination of parameters is
Prediction: One use of a fitted regression equation is to predict response-variable values for particular ‘‘future’’ combinations of explanatory-variable scores. Suppose, therefore, that we fit the model y ¼ Xfl þ ", obtaining the least-squares estimate b of fl. Let x0 0 ¼ ½1; x01; ...
Suppose that the model matrix for the two-way ANOVA model Yijk ¼ µ þ αj þ βk þ γjk þ εijk is reduced to full rank by imposing the following constraints (for r ¼ 2 rows and c ¼ 3 columns):α2 ¼ 0β3 ¼ 0γ21 ¼ γ22 ¼ γ13 ¼ γ23 ¼ 0 These constraints imply dummy-variable (0/1)
*Show that the equation-by-equation least-squares estimator Bb ¼ ðX0 XÞ(1 X0 Yis the maximum-likelihood estimator of the regression coefficients B in the multivariate general linear model Y ¼ XB þ E, where the model matrix X is fixed, and the distribution of the errors is "i ; Nmð0; SÞ, with
Intention to treat: Recall the imaginary example in Section 9.8.1 in which students were randomly provided vouchers to defray the cost of attending a private school. In the text, we imagined that the researchers want to determine the effect of private versus public school attendance on academic
*The asymptotic covariance matrix of the IV estimator is76 V ¼ 1 nplim nð Þ bIV ( fl ð Þ bIV ( fl 0 * +The IV estimator itself (Equation 9.28) can be written as bIV ¼ fl þ ðZ0 XÞ(1 Z0 "(Reader: Why?) Then, V ¼ 1 n plim nðZ0 XÞ(1 Z0 ""0 ZðX0 ZÞ(1 h i Starting with this formula, show
Show that when the model matrix X is used as the IV matrix Z in instrumentalvariables estimation, the IV and OLS estimators and their covariance matrices coincide. See Equations 9.28 and 9.29 (on page 233).
Two-stage least-squares estimation:(a) Suppose that the column x1 in the model matrix X also appears in the matrix Z of instrumental variables in 2SLS estimation. Explain why bx1 in the first-stage regression simply reproduces x1; that is, bx1 ¼ x1.(b) *Show that the two formulas for the 2SLS
+The second principal component is w2ðn · 1Þ¼ A12z1 þ A22z2 þ###þ Ak2zk¼ ZXðn · kÞa2ðk · 1Þwith variance S2 W2 ¼ a0 2RXX a2 We need to maximize this variance subject to the normalizing constraint a0 2a2 ¼ 1 and the orthogonality constraint w0 1w2 ¼ 0. Show that the orthogonality
+Find the matrix A of principal-component coefficients when k ¼ 2 and r12 is negative. (Cf. Equation 13.6 on page 352.)
+Show that when k ¼ 2, the principal components of RXX correspond to the principal axes of the data ellipse for the standardized regressors Z1 and Z2; show that the halflength of each axis is equal to the square root of the corresponding eigenvalue of RXX . Now extend this reasoning to the
+Use the principal-components analysis of the explanatory variables in B. Fox’s time-series regression, given in Table 13.3, to estimate the nearly collinear relationships among the variables corresponding to small principal components. Which variables appear to be involved in each nearly
+Show that Equation 13.7 (page 358) applied to the correlation matrix of the least-squares regression coefficients, computed from the coefficient covariance matrix S2 EðX0 XÞ'1, produces the same generalized variance-inflation factor as when it is applied to the correlation matrix of the Xs.
Why are there 2k ' 1 distinct subsets of k explanatory variables? Evaluate this quantity for k ¼ 2, 3; ... ; 15.
Apply the backward, forward, and forward/backward stepwise regression methods to B. Fox’s Canadian women’s labor force participation data. Compare the results of these procedures with those shown in Figure 13.8, based on the application of the BIC to all subsets of predictors.
+Show that the ridge-regression estimator of the standardized regression coefficients, b+d ¼ ðRXX þ dIk Þ'1 rXy can be written as a linear transformation b+d ¼ Ub+ of the usual least-squares estimator b+ ¼ R'1 XX rXy, where the transformation matrix is U [ Ik þ dR'1 XX $ %'1.
+Show that the variance of the ridge estimator is Vðb+dÞ ¼ σ+2εn ' 1ðRXX þ dIk Þ'1 RXX ðRXX þ dIk Þ'1[Hint: Express the ridge estimator as a linear transformation of the standardized response variable, b+d ¼ ðRXX þ dIk Þ'1½1=ðn ' 1Þ)Z0 X zy.]
+Finding the ridge constant d: Hoerl and Kennard suggest plotting the entries in b+d against values of d ranging between 0 and 1. The resulting graph, called a ridge trace, both furnishes a visual representation of the instability due to collinearity and (ostensibly) provides a basis for selecting
Vary the span of the kernel estimator for the regression of prestige on income in the Canadian occupational prestige data. Does s ¼ 0:4 appear to be a reasonable choice?
Selecting the span by smoothing residuals: A complementary visual approach to selecting the span in local-polynomial regression is to find the residuals from the fit from the local regression, Ei ¼ Yi & Ybi, and to smooth the residuals against the xi. If the data have been oversmoothed, then there
Comparing the kernel and local-linear estimators: To illustrate the reduced bias of the local-linear estimator in comparison to the kernel estimator, generate n ¼ 100 observations of artificial data according to the cubic regression equation Y ¼ 100 & 5 x10 & 5( ) þx 10 & 5( )3þ ε
*Bias, variance, and MSE as a function of bandwidth: Consider the artificial regression function introduced in the preceding exercise. Using Equation 18.1 (page 537), write down expressions for the expected value and variance of the local-linear estimator as a function of the bandwidth h of the
*Employing the artificial data generated in Exercise 18.3, use Equation 18.3 (on page 539) to compute the average squared error (ASE) of the local-linear regression estimator for various spans between s ¼ 0:05 and s ¼ 0:95, drawing a graph of ASEðsÞ versus s. What span produces the smallest
*Continuing with the artificial data from Exercise (8.3), graph the crossvalidation function CVðsÞ and generalized cross-validation function GCVðsÞ as a function of span, letting the span range between s ¼ 0:05 and s ¼ 0:95.(a) Compare the shape of CVðsÞ with the average squared error
Comparing polynomial and local regression:(a) The local-linear regression of prestige on income with span s ¼ 0:6 (in Figure 18.7 on page 543) has 5:006 equivalent degrees of freedom, very close to the number of degrees of freedom for a global fourth-order polynomial. Fit a fourth-order polynomial
Equivalent kernels: One way of comparing linear smoothers like local-polynomial estimators and smoothing splines is to think of them as variants of the kernel estimator, where fitted values arise as weighted averages of observed response values. This approach is illustrated in Figure 18.19, which
*Prove that the median minimizes the least-absolute-values objective function:Xn i¼1 rLAVðEiÞ ¼ Xn i¼1 j j Yi % µb
Breakdown: Consider the contrived data set Y1 ¼ %0:068 Y2 ¼ %1:282 Y3 ¼ 0:013 Y4 ¼ 0:141 Y5 ¼ %0:980(an adaptation of the data used to construct Figure 19.1). Show that more than two values must be changed to influence the median of the five values to an arbitrary degree. (Try, e.g., to make
The following contrived data set (discussed in Chapter 3) is from Anscombe(1973):(a) Graph the data and confirm that the third observation is an outlier. Find the leastsquares regression of Y on X, and plot the least-squares line on the graph.(b) Fit a robust regression to the data using the
Computing the LTS estimator: Why is it almost surely the case that theðk þ 1Þ · ðk þ 1Þ matrix X*, with rows selected from among those of the complete model matrix X, is of full rank when all its rows are different? (Put another way, how is it possible that X* would not be of full rank?)
In Chapter 15, I fit a Poisson regression of number of interlocks on assets, nation of control, and sector for Ornstein’s Canadian interlocking-directorate data. The results from this regression are given in Table 15.3 (page 428). Influential-data diagnostics (see, e.g., Figure 15.7 on page 456)
Consider the following contrived data set for the variables X1, X2, and X3, where the question marks indicate missing data:(a) Using available cases (and recomputing the means and standard deviations for each pair of variables), find the pairwise correlations among the three variables and explain
'In univariate missing data, where there are missing values for only one variable in a data set, some of the apparently distinct methods for handling missing data produce identical results for certain statistics. Consider Table 20.1 on page 612, for example, where data are missing on the variable
'Duplicate the small simulation study reported in Table 20.2 on page 613, comparing several methods of handling univariate missing data that are MAR. Then repeat the study for missing data that are MCAR and for missing data that are MNAR (generated as in Figure 20.1 on page 608). What do you
'Equation 20.6 (on page 616) gives the ML estimators for the parameters µ1,µ2, σ2 1, σ2 2, and σ12 in the bivariate-normal model with some observations on X2 missing at random but X1 completely observed. The interpretation of µb1 and σb2 1 is straightforward: They are the available-case mean
'Multivariate linear regression fits the model Yðn · mÞ¼ Xðn · kþ1ÞBðkþ1 · mÞþ Eðn · mÞwhere Y is a matrix of response variables; X is a model matrix (just as in the univariate linear model); B is a matrix of regression coefficients, one column per response variable; and E is a
'Consider once again the case of univariate missing data MAR for two bivariately normal variables, where the first variable, X1; is completely observed, and m observations(for convenience, the first m) on the second variable, X2, are missing.(a) Let A'2j1 and B'2j1 represent the intercept and slope
As explained in Section 20.4.1, the efficiency of the multiple-imputation estimator of a coefficient βej relative to the ML estimator bβj is REðβejÞ ¼ g=ðg þ γjÞ, where g is the number of imputations employed and γj is the rate of missing information for coefficientβj. The square root
Examine the United Nations data on infant mortality and other variables for 207 countries, discussed in Section 20.4.4.(a) Perform a complete-case linear least-squares regression of infant mortality on GDP per capita, percentage using contraception, and female education. Does it appear reasonable
Truncated normal distributions:(a) Suppose that j ; Nð0; 1Þ. Using Equations 20.14 (page 630) for the mean and variance of a left-truncated normal distribution, calculate the mean and variance of j j j > a for each of a ¼ &2; &1; 0; 1; and 2.(b) 'Find similar formulas for the mean and variance
'Suppose that j ; Nðµ; σ2Þ is left-censored at j ¼a, so that Y ¼ a for j £ a j for j > a, Using Equations 20.14 (on page 630) for the truncated normal distribution, show that (repeating Equations 20.16 on page 631)EðYÞ ¼ aFðzaÞ þ ½ ) µ þ σmðzaÞ ½ ) 1 & FðzaÞVðYÞ ¼ σ2½ ) 1
'Equations 20.16 (on page 631) give formulas for the mean and variance of a left-censored normally distributed variable. (These formulas are also shown in the preceding exercise.) Derive similar formulas for (a) a right-censored and (b) an interval-censored normally distributed variable.
'Using Equations 20.17 (page 631) for the incidentally truncated bivariate-normal distribution, show that the expectation of the error εi in the Heckman regression model(Equations 20.18 and 20.19 on page 632) conditional on Y being observed is Eðεi j ζi > 0Þ ¼ Eðεijδi > & ciÞ ¼
'As explained in the text, the Heckman regression model (Equations 20.18 and 20.19, page 632) implies thatðYijζi > 0Þ ¼ α þ β1Xi1 þ β2Xi2 þ***þ βkXik þ βλλi þ ni where βλ [ σεrεδ, λi [ m &ci ð Þ, and ci ¼ γ0 þ γ1Zi1 þ γ2Zi2 þ***þ γpZip Show that the errors ni are
'The log-likelihood for the Heckman regression-selection model is given in Equation 20.20 (page 634). Derive this expression. (Hint: The first sum in the log-likelihood, for the observations for which Y is missing, is of the log-probability that each such Yi is missing; the second sum is of the log
'Explain how White’s coefficient-variance estimator (see Section 12.2.3), which is used to correct the covariance matrix of OLS regression coefficients for heteroscedasticity, can be employed to obtain consistent coefficient standard errors for the two-step estimator of Heckman’s
Greene (2003 p. 768) remarks that the ML estimates βbj of the regression coefficients in a censored-regression model are often approximately equal to the OLS estimates Bj divided by the proportion P of uncensored observations; that is, βbj » Bj=P. Does this pattern hold for the hours-worked
'Test the omnibus null hypothesis H0: β1 ¼ β2 ¼ 0 for the Huber M estimator in Duncan’s regression of occupational prestige on income and education.(a) Base the test on the estimated asymptotic covariance matrix of the coefficients.(b) Use the bootstrap approach described in Section 21.4.
Case weights:(a) 'Show how case weights can be used to ‘‘adjust’’ the usual formulas for the leastsquares coefficients and their covariance matrix. How do these case-weighted formulas compare with those for weighted-least-squares regression (discussed in Section 12.2.2.)?(b) Using data from
'Bootstrapping time-series regression: Bootstrapping can be adapted to timeseries regression but, as in the case of fixed-X resampling, the procedure makes strong use of the model fit to the data—in particular, the manner in which serial dependency in the data is modeled. Suppose that the errors
'Prove that Mallows’s Cp statistic, Cpj ¼ RSSj S2 Eþ 2sj # n can also be written Cpj ¼ ðk þ 1 # sjÞðFj # 1Þ þ sj where RSSj is the residual sum of squares for model Mj; sj is the number of parameters (including the constant) in model Mj; n is the number of observations; S2 E is the usual
Both the adjusted R2, Re2 ¼ 1 # n # 1 n # s·RSS TSS and the generalized cross-validation criterion GCV ¼ n · RSS ð Þ n # s 2 penalize models that have large numbers of predictors. (Here, n is the number of observations, s the number of parameters in the model, RSS the residual sum of squares
Show that the differences in BIC values given in the first column of Table 22.1(page 680) correspond roughly to the Bayes factors and posterior model probabilities given in columns 2 and 3 of the table.
Perform model selection for the baseball salary regression using a criterion or criteria different from the BIC, examining the ‘‘best’’ model of each size, and the ‘‘best’’ 10 or 15 models regardless of size. Are the models similar to those nominated by the BIC? Why did you obtain
Using the estimated fixed effects in the table on page 717 for the model fit to the High School and Beyond data, find the fixed-effect regression equations for typical low, medium, and high mean SES Catholic and public schools, as plotted in Figure 23.6.
*BLUPs: As discussed in Section 23.8, show that for the random-effects oneway ANOVA model, Yij ¼ β1 þ δ1i þ εij, the weights wi ¼ ni minimize the variance of the estimatorβb1 ¼Pm Pi¼1 wiYi#m i¼1 wi and thus that this choice provides the best linear unbiased estimator (BLUE) of β1. Then
*Prove that the least-squares estimates of the coefficient β2 for Xij is the same in the following two fixed-effects models (numbered as in Section 23.7.1):Recall the context: The data are divided into groups i ¼ 1; ... ; m, with individuals j ¼ 1; ... ; ni in the ith group. The first model
*Using V –"( ) ¼ c* 0 0 σ2εL( )show that the covariance matrix of the response variable in the compact form of the LMM, y ¼ Xfl þ Z– þ ", can be written as VðyÞ ¼ Zc*Z0 þ σ2εL.58
*Show that the log-likelihood for the variance-covariance-component parameters ! given the fixed effects fl can be written as (repeating Equation 23.22 from page 737)loge Lð!jfl; yÞ¼& n 2logeð2πÞ & 1 2loge½ )& det Qð!Þ 1 2ðy & XflÞQ&1ð!Þðy & XflÞ
Further on migraine headaches:(a) A graph of the fixed effects for the mixed-effects logit model fit to the migraine headaches data is shown in Figure 24.1 (page 747), and the estimated parameters of the model are given on page 746. Explain how the lines on the graph, showing how the fitted
Further on recovery from coma:(a) The example in Section 24.2.1 on recovery from coma uses data on performance IQ.The original analysis of the data by Wong et al. (2001) also examined verbal IQ. Repeat the analysis using verbal IQ as the response variable, employing the nonlinear mixed-effects
'Show that the correlation between the least-squares residuals Ei and the response-variable values Yi is 1 " R2 p . [Hint: Use the geometric vector representation of multiple regression (developed in Chapter 10), examining the plane in which the e; y'; and by ' vectors lie.]
Nonconstant variance and specification error: Generate 100 observations according to the following model:Y ¼ 10 þ ð1 · XÞþð1 · DÞþð2 · X · DÞ þ εwhere ε ; Nð0; 102Þ; the values of X are 1; 2; ... ; 50; 1; 2; ... ; 50; the first 50 values of D are 0; and the last 50 values of D
'Weighted-least-squares estimation: Suppose that the errors from the linear regression model y ¼ X fl þ " are independent and normally distributed, but with different variances, εi ; Nð0; σ2 i Þ, and that σ2 i ¼ σ2ε=w2 i . Show that:(a) The likelihood for the model is Lðfl; σ2ε Þ ¼
'Show that when the covariance matrix of the errors is S ¼ σ2ε · diagf1=W2 1 ; ... ; 1=W2 n g [ σ2εW"1 the weighted-least-squares estimator flb ¼ ðX0 WXÞ"1 X0 Wy¼ My is the minimum-variance linear unbiased estimator of fl (Hint: Adapt the proof of the GaussMarkov theorem for OLS
'The impact of nonconstant error variance on OLS estimation: Suppose that Yi ¼ α þ βxi þ εi, with independent errors, εi ; Nð0; σ2 i Þ, and σi ¼ σεxi. Let B represent the OLS estimator and bβ the WLS estimator of β.(a) Show that the sampling variance of the OLS estimator is VðBÞ
Experimenting with component-plus-residual plots: Generate random samples of 100 observations according to each of the following schemes. In each case, construct the component-plus-residual plots for X1 and X2. Do these plots accurately capture the partial relationships between Y and each of X1 and
Consider an alternative analysis of the SLID data in which log wages is regressed on sex, transformed education, and transformed age—that is, try to straighten the relationship between log wages and age by a transformation rather than by a quadratic regression. How successful is this approach?
Apply Mallows’s procedure to construct augmented component-plus-residual plots for the SLID regression of log wages on sex, age, and education. *Then apply Cook’s CERES procedure to this regression. Compare the results of these two procedures with each other and with the ordinary
*Figure 2.7 illustrates how, when the relationship between Y and X is nonlinear in an interval, the average value of Y in the interval can be a biased estimate of EðYjxÞat the center of the interval. Imagine that X-values are evenly distributed in an interval centered at xi, and let µi ”
Create a graph like Figure 4.1, but for the ordinary power transformations X ! Xp for p ¼ #1; 0; 1; 2; 3. (When p ¼ 0, however, use the log transformation.) Compare your graph to Figure 4.1, and comment on the similarities and differences between the two families of transformations Xp and XðpÞ.
&Show that the derivative of f ðXÞ¼ðXp # 1Þ=p is equal to 1 at X ¼ 1 regardless of the value of p.
&We considered starts for transformations informally to ensure that all data values are positive and that the ratio of the largest to the smallest data values is sufficiently large.An alternative is to think of the start as a parameter to be estimated along with the transformation power to make the
The Yeo-Johnson family of modified power transformations (Yeo & Johnson, 2000) is an alternative to using a start when both negative (or 0) and positive values are included in the data. The Yeo-Johnson family is defined as follows:X ! X½p)[ ðX þ 1ÞðpÞ for X ‡ 0ð1 # XÞð2#pÞ for X <
'Prove that the least-squares fit in simple-regression analysis has the following properties:(a) P YbiEi ¼ 0.(b) PðYi & YbiÞðYbi & YÞ ¼ PEiðYbi & YÞ ¼ 0.
'Suppose that the means and standard deviations of Y and X are the same:Y ¼ X and SY ¼ SX .(a) Show that, under these circumstances, BYjX ¼ BX jY ¼ rXY where BYjX is the least-squares slope for the simple regression of Y on X, BXjY is the least-squares slope for the simple regression of X on Y,
'Show that A0 ¼ Y minimizes the sum of squares SðA0Þ ¼ Xn i¼1ðYi & A0Þ2
Linear transformation of X and Y:(a) Suppose that the explanatory-variable values in Davis’s regression are transformed according to the equation X0 ¼ X & 10 and that Y is regressed on X0. Without redoing the regression calculations in detail, find A0 , B0 , S0 E, and r0 . What happens to these
'Derive the normal equations (Equations 5.7) for the least-squares coefficients of the general multiple-regression model with k explanatory variables. [Hint: Differentiate the sum-of-squares function SðA; B1; ... ; Bk Þ with respect to the regression coefficients, and set the partial derivatives
Why is it the case that the multiple-correlation coefficient R2 can never get smaller when an explanatory variable is added to the regression equation? [Hint: Recall that the regression equation is fit by minimizing the residual sum of squares, which is equivalent to maximizing R2 (why?).]
Consider the general multiple-regression equation Y ¼ A þ B1X1 þ B2X2 þ***þ BkXk þ E An alternative procedure for calculating the least-squares coefficient B1 is as follows:1. Regress Y on X2 through Xk , obtaining residuals EYj2 ... k .2. Regress X1 on X2 through Xk , obtaining residuals
Partial correlation: The partial correlation between X1 and Y ‘‘controlling for’’X2 through Xk is defined as the simple correlation between the residuals EYj2 ... k and E1j2 ... k , given in the previous exercise. The partial correlation is denoted rY1j2...k .(a) Using the Canadian
'Show that in simple-regression analysis, the standardized slope coefficient B' is equal to the correlation coefficient r. (In general, however, standardized slope coefficients are not correlations and can be outside of the range [0, 1].)
+Demonstrate the unbias of the least-squares estimators A and B of α and β in simple regression:(a) Expressing the least-squares slope B as a linear function of the observations, B ¼ PmiYi (as in the text), and using the assumption of linearity, EðYiÞ ¼ α þ βxi, show that EðBÞ ¼ β.
+Using the assumptions of linearity, constant variance, and independence, along with the fact that A and B can each be expressed as a linear function of the Yis, derive the sampling variances of A and B in simple regression. [Hint: VðBÞ ¼ Pm2 i VðYiÞ.]
Examining the formula for the sampling variance of A in simple regression, VðAÞ ¼ σ2εPx2 in Pðxi ' xÞ2 why is it intuitively sensible that the variance of A is large when the mean of the xs is far from 0? Illustrate your explanation with a graph.
The formula for the sampling variance of B in simple regression, VðBÞ ¼ σ2 P εðxi ' xÞ2 shows that, to estimate β precisely, it helps to have spread out xs. Explain why this result is intuitively sensible, illustrating your explanation with a graph. What happens to VðBÞ when there is no
+Maximum-likelihood estimation of the simple-regression model: Deriving the maximum-likelihood estimators of α and β in simple regression is straightforward. Under the assumptions of the model, the Yis are independently and normally distributed random variables with expectations α þ βxi and
Linear transformation of X and Y in simple regression (continuation of Exercise 5.4):(a) Suppose that the X-values in Davis’s regression of measured on reported weight are transformed according to the equation X0 ¼ 10ðX ' 1Þ and that Y is regressed on X0.Without redoing the regression
Consider the regression model Y ¼ α þ β1x1 þ β2x2 þ ε. How can the incremental sum-of-squares approach be used to test the hypothesis that the two population slopes are equal to each other, H0: β1 ¼ β2? [Hint: Under H0, the model becomes Y ¼ αþβx1 þβx2 þε ¼ Y ¼ αþβðx1
Examples of specification error (also see the discussion in Section 9.7):(a) Describe a nonexperimental research situation—real or contrived—in which failure to control statistically for an omitted variable induces a correlation between the error and an explanatory variable, producing erroneous
Suppose that the ‘‘true’’ model generating a set of data is Y ¼ α þ β1X1 þ ε, where the error ε conforms to the usual linear-regression assumptions. A researcher fits the model Y ¼ α þ β1X1 þ β2X2 þ ε, which includes the irrelevant explanatory variable X2—that is, the true
+Derive Equations 6.12 by multiplying Equation 6.11 through by each of X1 and X2. (Hints: Both X1 and X2 are uncorrelated with the regression error ε. Likewise, X2 is uncorrelated with the measurement error δ. Show that the covariance of X1 and δ is simply the measurement error variance σ2δ by
+Show that the population analogs of the regression coefficients can be written as in Equations 6.14. (Hint: Ignore the measurement errors, and derive the population analogs of the normal equations by multiplying the ‘‘model’’ Y ¼ β1X1 þ β2X2 þ ε through by each of X1 and X2 and
+Show that the variance of X1 ¼ τ þ δ can be written as the sum of ‘‘true-score variance,’’σ2τ , and measurement error variance, σ2δ . (Hint: Square both sides of Equation 6.10 and take expectations.)

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