{egin{tabular}{|c|c|c|c|}hline & A & B & C \hline 1 & multicolumn{3}{|c|}{ Downtown } \hline 2 & year & quarter & Profit \hline 3 & 2000 & 1 & $66,127.00 \hline 4 & 2000 & 2 & $52,652.00 \hline 5 & 2000 & 3 & $83,993.00 \hline 6 & 2000 & 4 & $48,053.00 \hline 7 & 2001 & 1 & $94,528.00 \hline 8 & 2001 & 2 & $66,254.00 \hline 9 & 2001 & 3 & $91,925.00 \hline 10 & 2001 & 4 & $88,179.00 \hline 11 & 2002 & 1 & $91,686.00 \hline 12 & 2002 & 2 & $86,379.00 \hline 13 & 2002 & 3 & $97,601.00 \hline 14 & 2002 & 4 & $56,077.00 \hline 15 & 2003 & 1 & $73,569.00 \hline 16 & 2003 & 2 & $73,167.00 \hline 17 & 2003 & 3 & $95,463.00 \hline 18 & 2003 & 4 & $91,694.00 \hline 19 & 2004 & 1 & $33,037.00 \hline 20 & 2004 & 2 & $38,051.00 \hline 21 & 2004 & 3 & -$745.00 \hline 22 & 2004 & 4 & $20,989.00 \hline 23 & 2005 & 1 & $53,098.00 \hline 24 & 2005 & 2 & $12,199.00 \hline 25 & 2005 & 3 & $47,069.00 \hline 26 & 2005 & 4 & $23,893.00 \hline 27 & 2006 & 1 & $89,081.00 \hline 28 & 2006 & 2 & $85,672.00 \hline 29 & 2006 & 3 & $97,209.00 \hline 30 & 2006 & 4 & $87,871.00 \hline 31 & 2007 & 1 & $82,944.00 \hline 32 & 2007 & 2 & $95,684.00 \hline 33 & 2007 & 3 & $57,783.00 \hline 34 & 2007 & 4 & $66,139.00 \hline 35 & 2008 & 1 & $84,055.00 \hline 36 & 2008 & 2 & $83,187.00 \hline 37 & 2008 & 3 & $48,328.00 \hline 38 & 2008 & 4 & $34,459.00 \hline 39 & 2009 & 1 & $20,974.00 \hline 40 & 2009 & 2 & $6,067.00 \hline 41 & 2009 & 3 & $10,893.00 \hline 42 & 2009 & 4 & -$14,384.00 \hline 43 & 2010 & 1 & $126,175.00 \hline 44 & 2010 & 2 & $116,575.00 \hline 45 & 2010 & 3 & $86,858.00 \hline 46 & 2010 & 4 & $115,579.00 \hline 47 & 2011 & 1 & $87,419.00 \hline 48 & 2011 & 2 & $71,169.00 \hline 49 & 2011 & 3 & $57,851.00 \hline 50 & 2011 & 4 & $68,185.00 \hline 51 & 2012 & 1 & $114,982.00 \hline 52 & 2012 & 2 & $84,497.00 \hline 53 & 2012 & 3 & $92,424.00 \hline 54 & 2012 & 4 & $89,340.00 \hline 55 & 2013 & 1 & $2,170.00 \hline 56 & 2013 & 2 & $18,114.00 \hline 57 & 2013 & 3 & $18,045.00 \hline 58 & 2013 & 4 & $4,536.00 \hline 59 & 2014 & 1 & $63,143.00 \hline 60 & 2014 & 2 & $79,762.00 \hline 61 & 2014 & 3 & $80,371.00 \hline 62 & 2014 & 4 & $67,942.00 \hline end{tabular}}
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39. Antigenic evolution and vaccination Antigenic data like those in Figure 14 can be summarized by taking the points from each year and drawing the smallest possible circle that encompasses these data. This results in a temporal sequence of circles in antigenic space, one for each year. Similarly, if the antigenic data are in three dimensions, spheres can be drawn around the data for each year,as shown in the figure. If the circles (or spheres) from years$x$ and $x+1$ overlap, then the amount of antigenic change between these years is relatively small. In such cases we might expect a single vaccine to work for both years. If the circles or spheres do not overlap, then we might need different vaccines for each year. (a)Suppose the antigenic data are two-dimensional,and the circles for two successive years are givenby the equations $(x-2)^{2}+(y-3)^{2}=1$ and $(x-3)^{2}+(y-2)^{2}=\frac{1}{4} .$ Would a single vaccine work for both vears? (b) Suppose the antigenic data are three-dimensional, and the spheres for two successive years are given by the equations and $$\begin{aligned}(x-2)^{2}+(y-3)^{2}+(z-1)^{2} &=1 \\(x-3)^{2}+(y-2)^{2}+z^{2} &=\frac{1}{4} \end{aligned}$$ Would a single vaccine work for both years? (c) Notice that the x- and y-coordinates of the centers of the circles in part (a) are the same as the x- and y-coordinates of the centers of the spheres in part (b), and the radii are the same as well. What is the relationship between the plot of the circles in part (a) and the plot of the spheres in part (b)?
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This question asks you to study the so-called Beveridge Curve from the perspective of cointegration analysis. The U.S. monthly data from December 2000 through February 2012 are in BEVERIDGE.RAW. (i) Test for a unit root in urate using the usual Dickey-Fuller test (with a constant) and the augmented DF with two lags of curate. What do you conclude? Are the lags of curate in the augmented DF test statistically significant? Does it matter to the outcome of the unit root test? (ii) Repeat part (i) but with the vacancy rate, vrate. (iii) Assuming that urate and vrate are both I(1), the Beveridge curve, $$u r a t e_{t}=\alpha+\beta vrate +u_{t}$$ only makes sense if urate and vrate are cointegrated (with cointegrating parameter $\beta<0 )$ . Test for cointegration using the Engle-Granger test with no lags. Are urate and vrate cointegrated at the 10$\%$ significance level? What about at the 5$\%$ level? (iv) Obtain the leads and lags estimator with $cvrate_{t}$, $cvrate_{t-1}$ and $cvrate_{t+1}$ as the I(O) explanatory variables added to the equation in part (ii). Obtain the Newey- West standard error for $\hat{\beta}$ using four lags $(\mathrm{so} g=4$ in the notation of Section 12.5$) .$ What is the resulting 95$\%$ confidence interval for $\beta$ How does it compare with the confidence interval that is not robust to serial correlation (or heteroskedasticity)? (v) Redo the Engle-Granger test but with two lags in the augmented DF regression. What happens? What do you conclude about the robustness of the claim that urate and vrate are cointegrated?
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