Question 5. (30 points) The 'business cycle' is not a regular cycle like a sine function. It refers to the fact that the economy as a whole has ups and downs. We can see this quite easily by looking at a measure of the total economy such as the GDP. Many aspects

of the economy commonly tend to move more-or-less together, such as employment, retail sales, the stock market, etc. But we know from many decades of evidence across many countries that they do not all move one-for-one. Some aspects of the economy tend to go up and down more than others. They are more volatile and the standard deviations of their growth rates are larger. Let's use some real data to check this out. We will use data from FRED: https://fred. stlouisfed.org/ They also have an Excel add-in https://fred.stlouisfed.org/fred-addin/ along with a user guide. You can use that add-in or you can use their menus in your browser to get the necessary data. The variables you want are Variable Name FRED Code GDP GDPC1 Consumption PCEC Investment GPDI If you are not using the Excel add-in, you can also get these variables the 'long way'. 1) Go to FRED, 2) Type GDPC1 in the search box in the upper right hand corner of the web page. This gives you a link. 3) click on that link and you get a graph, 4) click on the col- orful button 'EDIT GRAPH', 5) Choose 'ADD LINE', 6) Enter the code for Consumption, 7) click on the add data series button, 8) Repeat this process for Investment. You should now see a graph with the three colored lines. It also shows vertical grey bars which show recessions. 9) In the upper right hand corner of the page you should see a 'DOWNLOAD' button which allows you to download in various formats. Pick the Excel data format so that you can use the data. Save it somewhere on you computer that you can find! You should now have the data in a nice Excel file on your computer. Using this data, in Excel for each series compute the discretely-compounded rates of growth over the period 1950Q1 to 2017Q4 inclusive. The formula you want is: g_{t= (}x_t x_t1 /x_t1)100 = (x_t/x_t11)100 Notice that the 100 at the end converts the quarterly growth rate into a percentage. No- tice that in order to get the first time period growth rate you need data from 1949Q4. Put the three growth rates into their own columns. For each of the following questions you will do the calculations and plotting once in Excel. You will submit your written answers in a Word file. You will also submit the Excel file that justify what you wrote in the Word file.

1. For each of the growth rates calculate the mean and standard deviation. How do the standard deviations compare to each other? 2. What are the correlations among the three values? Do you agree with the claim that the variables 'move up and down together but have different volatilities'? Use a graph to illustrate your answer. 3. Calculate the continuously compounded growth rate of GDP. The formula you want is t = (ln xt ln xt1) 100 How do the mean and standard deviation compare to those calculated with the discretely compounded growth rate? Why are they so similar?

GDPC1	Real Gross Domestic Product, Billions of Chained 2017 Dollars, Quarterly, Seasonally Adjusted Annual Rate
DPCCRV1Q225SBEA	Personal Consumption Expenditures (PCE) Excluding Food and Energy (Chain-Type Price Index), Percent Change from Preceding Period, Quarterly, Seasonally Adjusted Annual Rate
DGS10	Market Yield on U.S. Treasury Securities at 10-Year Constant Maturity, Quoted on an Investment Basis, Percent, Daily, Not Seasonally Adjusted
Frequency: Quarterly				Frequency: Daily
observation_date	GDPC1	DPCCRV1Q225SBEA		observation_date	DGS10
1947-01-01	2182.681			1962-01-02	4.06
1947-04-01	2176.892			1962-01-03	4.03
1947-07-01	2172.432			1962-01-04	3.99
1947-10-01	2206.452			1962-01-05	4.02
1948-01-01	2239.682			1962-01-08	4.03
1948-04-01	2276.690			1962-01-09	4.05
1948-07-01	2289.770			1962-01-10	4.07
1948-10-01	2292.364			1962-01-11	4.08
1949-01-01	2260.807			1962-01-12	4.08
1949-04-01	2253.128			1962-01-15	4.10
1949-07-01	2276.424			1962-01-16	4.13
1949-10-01	2257.352			1962-01-17	4.12
1950-01-01	2346.104			1962-01-18	4.11
1950-04-01	2417.682			1962-01-19	4.11
1950-07-01	2511.127			1962-01-22	4.09
1950-10-01	2559.214			1962-01-23	4.11
1951-01-01	2593.967			1962-01-24	4.10
1951-04-01	2638.898			1962-01-25	4.11
1951-07-01	2693.259			1962-01-26	4.11
1951-10-01	2699.156			1962-01-29	4.12
1952-01-01	2727.954			1962-01-30	4.11
1952-04-01	2733.800			1962-01-31	4.10
1952-07-01	2753.517			1962-02-01	4.09
1952-10-01	2843.941			1962-02-02	4.08
1953-01-01	2896.811			1962-02-05	4.07
1953-04-01	2919.206			1962-02-06	4.06
1953-07-01	2902.785			1962-02-07	4.07
1953-10-01	2858.845			1962-02-08	4.07
1954-01-01	2845.192			1962-02-09	4.05
1954-04-01	2848.305			1962-02-12	#N/A
1954-07-01	2880.482			1962-02-13	4.03
1954-10-01	2936.852			1962-02-14	4.03
1955-01-01	3020.746			1962-02-15	4.02
1955-04-01	3069.910			1962-02-16	4.02
1955-07-01	3111.379			1962-02-19	4.01
1955-10-01	3130.068			1962-02-20	4.05
1956-01-01	3117.922			1962-02-21	4.03
1956-04-01	3143.694			1962-02-22	#N/A
1956-07-01	3140.874			1962-02-23	4.02
1956-10-01	3192.570			1962-02-26	4.00
1957-01-01	3213.011			1962-02-27	4.01
1957-04-01	3205.970			1962-02-28	4.00
1957-07-01	3237.386			1962-03-01	3.98
1957-10-01	3203.894			1962-03-02	3.98
1958-01-01	3120.724			1962-03-05	4.00
1958-04-01	3141.224			1962-03-06	4.01
1958-07-01	3213.884			1962-03-07	4.00
1958-10-01	3289.032			1962-03-08	3.98
1959-01-01	3352.129			1962-03-09	3.96
1959-04-01	3427.667	2.2		1962-03-12	3.94
1959-07-01	3430.057	2.7		1962-03-13	3.92
1959-10-01	3439.832	2.3		1962-03-14	3.93
1960-01-01	3517.181	1.2		1962-03-15	3.96
1960-04-01	3498.246	1.5		1962-03-16	3.96
1960-07-01	3515.385	1.6		1962-03-19	3.93
1960-10-01	3470.278	1.3		1962-03-20	3.91
1961-01-01	3493.703	0.7		1962-03-21	3.86
1961-04-01	3553.021	1.3		1962-03-22	3.83
1961-07-01	3621.252	1.8		1962-03-23	3.87
1961-10-01	3692.289	1.0		1962-03-26	3.89
1962-01-01	3758.147	1.5		1962-03-27	3.90
1962-04-01	3792.149	1.5		1962-03-28	3.90
1962-07-01	3838.776	1.3		1962-03-29	3.90
1962-10-01	3851.421	0.8		1962-03-30	3.86
1963-01-01	3893.482	1.3		1962-04-02	3.86
1963-04-01	3937.183	1.5		1962-04-03	3.83
1963-07-01	4023.755	1.6		1962-04-04	3.78
1963-10-01	4050.147	1.9		1962-04-05	3.84
1964-01-01	4135.553	1.7		1962-04-06	3.81
1964-04-01	4180.592	1.2		1962-04-09	3.83
1964-07-01	4245.918	1.1		1962-04-10	3.83
1964-10-01	4259.046	1.0		1962-04-11	3.87
1965-01-01	4362.111	1.5		1962-04-12	3.86
1965-04-01	4417.225	1.2		1962-04-13	3.85
1965-07-01	4515.427	1.3		1962-04-16	3.83
1965-10-01	4619.458	1.3		1962-04-17	3.82
1966-01-01	4731.888	2.0		1962-04-18	3.81
1966-04-01	4748.046	3.3		1962-04-19	3.82
1966-07-01	4788.254	3.3		1962-04-20	#N/A
1966-10-01	4827.537	3.5		1962-04-23	3.86
1967-01-01	4870.299	2.2		1962-04-24	3.87
1967-04-01	4873.287	2.7		1962-04-25	3.88
1967-07-01	4919.392	3.7		1962-04-26	3.89
1967-10-01	4956.477	4.1		1962-04-27	3.86
1968-01-01	5057.553	4.6		1962-04-30	3.86
1968-04-01	5142.033	4.7		1962-05-01	3.85
1968-07-01	5181.859	4.5		1962-05-02	3.86
1968-10-01	5202.212	4.7		1962-05-03	3.86
1969-01-01	5283.597	4.4		1962-05-04	3.87
1969-04-01	5299.625	4.9		1962-05-07	3.86
1969-07-01	5334.600	4.7		1962-05-08	3.85
1969-10-01	5308.556	4.8		1962-05-09	3.82
1970-01-01	5300.652	4.4		1962-05-10	3.83
1970-04-01	5308.164	4.6		1962-05-11	3.82
1970-07-01	5357.077	4.5		1962-05-14	3.85
1970-10-01	5299.672	5.8		1962-05-15	3.87
1971-01-01	5443.619	5.2		1962-05-16	3.86
1971-04-01	5473.059	4.6		1962-05-17	3.87
1971-07-01	5518.072	3.8		1962-05-18	3.93
1971-10-01	5531.032	2.3		1962-05-21	3.94
1972-01-01	5632.649	3.9		1962-05-22	3.93
1972-04-01	5760.470	2.7		1962-05-23	3.91
1972-07-01	5814.854	3.0		1962-05-24	3.90
1972-10-01	5912.220	2.5		1962-05-25	3.88
1973-01-01	6058.544	2.8		1962-05-28	3.87
1973-04-01	6124.506	5.6		1962-05-29	3.89
1973-07-01	6092.301	5.4		1962-05-30	#N/A
1973-10-01	6150.131	5.7		1962-05-31	3.90
1974-01-01	6097.258	7.0		1962-06-01	3.89
1974-04-01	6111.751	10.3		1962-06-04	3.88
1974-07-01	6053.978	11.9		1962-06-05	3.89
1974-10-01	6030.464	10.1		1962-06-06	3.89
1975-01-01	5957.035	8.1		1962-06-07	3.87
1975-04-01	5999.610	6.2		1962-06-08	3.86
1975-07-01	6102.326	6.1		1962-06-11	3.87
1975-10-01	6184.530	6.7		1962-06-12	3.86
1976-01-01	6323.649	6.2		1962-06-13	3.87
1976-04-01	6370.025	4.9		1962-06-14	3.90
1976-07-01	6404.895	6.4		1962-06-15	3.89
1976-10-01	6451.177	6.4		1962-06-18	3.91
1977-01-01	6527.703	6.9		1962-06-19	3.93
1977-04-01	6654.466	6.3		1962-06-20	3.92
1977-07-01	6774.457	6.8		1962-06-21	3.90
1977-10-01	6774.592	5.9		1962-06-22	3.90
1978-01-01	6796.260	6.4		1962-06-25	3.95
1978-04-01	7058.920	7.1		1962-06-26	3.97
1978-07-01	7129.915	7.0		1962-06-27	3.96
1978-10-01	7225.750	7.4		1962-06-28	3.98
1979-01-01	7238.727	5.6		1962-06-29	4.00
1979-04-01	7246.454	9.1		1962-07-02	4.00
1979-07-01	7300.281	7.6		1962-07-03	4.00
1979-10-01	7318.535	8.6		1962-07-04	#N/A
1980-01-01	7341.557	10.2		1962-07-05	4.00
1980-04-01	7190.289	9.2		1962-07-06	4.02
1980-07-01	7181.743	9.2		1962-07-09	4.05
1980-10-01	7315.677	10.1		1962-07-10	4.02
1981-01-01	7459.022	9.2		1962-07-11	3.99
1981-04-01	7403.745	7.8		1962-07-12	4.00
1981-07-01	7492.405	7.6		1962-07-13	4.00
1981-10-01	7410.768	7.3		1962-07-16	4.01
1982-01-01	7295.631	6.0		1962-07-17	4.03
1982-04-01	7328.912	5.5		1962-07-18	4.02
1982-07-01	7300.896	6.6		1962-07-19	4.02
1982-10-01	7303.817	5.7		1962-07-20	4.01
1983-01-01	7400.066	5.4		1962-07-23	4.00
1983-04-01	7568.456	3.1		1962-07-24	4.01
1983-07-01	7719.746	6.2		1962-07-25	4.02
1983-10-01	7880.794	3.2		1962-07-26	4.01
1984-01-01	8034.847	4.1		1962-07-27	4.02
1984-04-01	8173.670	4.8		1962-07-30	4.03
1984-07-01	8252.465	3.8		1962-07-31	4.04
1984-10-01	8320.199	2.9		1962-08-01	4.03
1985-01-01	8400.820	5.6		1962-08-02	4.00
1985-04-01	8474.787	3.5		1962-08-03	4.01
1985-07-01	8604.220	4.2		1962-08-06	4.01
1985-10-01	8668.188	2.9		1962-08-07	4.02
1986-01-01	8749.127	4.4		1962-08-08	4.01
1986-04-01	8788.524	2.8		1962-08-09	3.99
1986-07-01	8872.601	2.8		1962-08-10	3.99
1986-10-01	8920.193	3.3		1962-08-13	3.99
1987-01-01	8986.367	2.6		1962-08-14	3.98
1987-04-01	9083.256	3.9		1962-08-15	3.98
1987-07-01	9162.024	3.6		1962-08-16	3.96
1987-10-01	9319.332	4.1		1962-08-17	3.96
1988-01-01	9367.502	3.9		1962-08-20	3.94
1988-04-01	9490.594	4.9		1962-08-21	3.94
1988-07-01	9546.206	4.7		1962-08-22	3.95
1988-10-01	9673.405	4.5		1962-08-23	3.95
1989-01-01	9771.725	4.4		1962-08-24	3.95
1989-04-01	9846.293	3.7		1962-08-27	3.95
1989-07-01	9919.228	3.0		1962-08-28	3.96
1989-10-01	9938.767	3.7		1962-08-29	3.96
1990-01-01	10047.386	4.7		1962-08-30	3.95
1990-04-01	10083.855	4.5		1962-08-31	3.96
1990-07-01	10090.569	4.1		1962-09-03	#N/A
1990-10-01	9998.704	3.4		1962-09-04	3.96
1991-01-01	9951.916	3.6		1962-09-05	3.97
1991-04-01	10029.510	3.1		1962-09-06	3.99
1991-07-01	10080.195	3.6		1962-09-07	4.01
1991-10-01	10115.329	3.2		1962-09-10	4.00
1992-01-01	10236.435	3.1		1962-09-11	4.00
1992-04-01	10347.429	2.9		1962-09-12	4.00
1992-07-01	10449.673	2.4		1962-09-13	3.99
1992-10-01	10558.648	3.0		1962-09-14	3.99
1993-01-01	10576.275	2.8		1962-09-17	3.99
1993-04-01	10637.847	3.1		1962-09-18	3.98
1993-07-01	10688.606	2.3		1962-09-19	3.99
1993-10-01	10833.987	2.1		1962-09-20	3.98
1994-01-01	10939.116	1.8		1962-09-21	3.98
1994-04-01	11087.361	2.7		1962-09-24	3.97
1994-07-01	11152.176	2.3		1962-09-25	3.96
1994-10-01	11279.932	2.0		1962-09-26	3.95
1995-01-01	11319.951	2.1		1962-09-27	3.95
1995-04-01	11353.721	2.4		1962-09-28	3.94
1995-07-01	11450.310	1.9		1962-10-01	3.93
1995-10-01	11528.067	2.0		1962-10-02	3.92
1996-01-01	11614.418	1.6		1962-10-03	3.90
1996-04-01	11808.140	1.9		1962-10-04	3.90
1996-07-01	11914.063	1.8		1962-10-05	3.92
1996-10-01	12037.775	2.3		1962-10-08	3.93
1997-01-01	12115.472	1.6		1962-10-09	3.95
1997-04-01	12317.221	2.1		1962-10-10	3.93
1997-07-01	12471.010	0.9		1962-10-11	3.94
1997-10-01	12577.495	1.3		1962-10-12	#N/A
1998-01-01	12703.742	1.2		1962-10-15	3.94
1998-04-01	12821.339	1.2		1962-10-16	3.94
1998-07-01	12982.752	1.4		1962-10-17	3.94
1998-10-01	13191.670	1.3		1962-10-18	3.92
1999-01-01	13315.597	1.0		1962-10-19	3.88
1999-04-01	13426.748	1.4		1962-10-22	3.91
1999-07-01	13604.771	1.5		1962-10-23	3.94
1999-10-01	13827.980	1.9		1962-10-24	3.94
2000-01-01	13878.147	2.1		1962-10-25	3.94
2000-04-01	14130.908	1.4		1962-10-26	3.93
2000-07-01	14145.312	1.9		1962-10-29	3.92
2000-10-01	14229.765	2.0		1962-10-30	3.92
2001-01-01	14183.120	2.6		1962-10-31	3.92
2001-04-01	14271.694	1.5		1962-11-01	3.92
2001-07-01	14214.516	1.0		1962-11-02	3.91
2001-10-01	14253.574	1.9		1962-11-05	3.91
2002-01-01	14372.785	1.2		1962-11-06	#N/A
2002-04-01	14460.848	2.2		1962-11-07	3.91
2002-07-01	14519.633	2.1		1962-11-08	3.90
2002-10-01	14537.580	1.4		1962-11-09	3.89
2003-01-01	14614.141	1.2		1962-11-12	#N/A
2003-04-01	14743.567	1.5		1962-11-13	3.90
2003-07-01	14988.782	1.8		1962-11-14	3.91
2003-10-01	15162.760	1.8		1962-11-15	3.91
2004-01-01	15248.680	2.2		1962-11-16	3.93
2004-04-01	15366.850	2.3		1962-11-19	3.93
2004-07-01	15512.619	1.4		1962-11-20	3.93
2004-10-01	15670.880	2.2		1962-11-21	3.92
2005-01-01	15844.727	2.9		1962-11-22	#N/A
2005-04-01	15922.782	2.0		1962-11-23	3.92
2005-07-01	16047.587	1.6		1962-11-26	3.93
2005-10-01	16136.734	2.7		1962-11-27	3.92
2006-01-01	16353.835	2.3		1962-11-28	3.92
2006-04-01	16396.151	3.2		1962-11-29	3.92
2006-07-01	16420.738	2.2		1962-11-30	3.92
2006-10-01	16561.866	1.7		1962-12-03	3.93
2007-01-01	16611.690	2.8		1962-12-04	3.94
2007-04-01	16713.314	1.7		1962-12-05	3.93
2007-07-01	16809.587	2.0		1962-12-06	3.92
2007-10-01	16915.191	2.8		1962-12-07	3.91
2008-01-01	16843.003	2.1		1962-12-10	3.90
2008-04-01	16943.291	1.7		1962-12-11	3.89
2008-07-01	16854.295	2.1		1962-12-12	3.88
2008-10-01	16485.350	-0.3		1962-12-13	3.88
2009-01-01	16298.262	0.0		1962-12-14	3.87
2009-04-01	16269.145	1.6		1962-12-17	3.85
2009-07-01	16326.281	1.4		1962-12-18	3.83
2009-10-01	16502.754	2.5		1962-12-19	3.84
2010-01-01	16582.710	1.4		1962-12-20	3.83
2010-04-01	16743.162	1.1		1962-12-21	3.81
2010-07-01	16872.266	0.6		1962-12-24	3.79
2010-10-01	16960.864	1.3		1962-12-25	#N/A
2011-01-01	16920.632	1.8		1962-12-26	3.79
2011-04-01	17035.114	2.4		1962-12-27	3.81
2011-07-01	17031.313	1.8		1962-12-28	3.84
2011-10-01	17222.583	1.4		1962-12-31	3.85
2012-01-01	17367.010	2.5		1963-01-01	#N/A
2012-04-01	17444.525	1.7		1963-01-02	3.82
2012-07-01	17469.650	1.2		1963-01-03	3.81
2012-10-01	17489.852	1.8		1963-01-04	3.84
2013-01-01	17662.400	1.6		1963-01-07	3.83
2013-04-01	17709.671	1.1		1963-01-08	3.81
2013-07-01	17860.450	1.6		1963-01-09	3.82
2013-10-01	18016.147	1.7		1963-01-10	3.81
2014-01-01	17953.974	1.3		1963-01-11	3.80
2014-04-01	18185.911	1.8		1963-01-14	3.81
2014-07-01	18406.941	1.5		1963-01-15	3.80
2014-10-01	18500.031	1.1		1963-01-16	3.80
2015-01-01	18666.621	0.7		1963-01-17	3.82
2015-04-01	18782.243	1.8		1963-01-18	3.84
2015-07-01	18857.418	1.3		1963-01-21	3.85
2015-10-01	18892.206	1.0		1963-01-22	3.86
2016-01-01	19001.690	1.7		1963-01-23	3.86
2016-04-01	19062.709	2.2		1963-01-24	3.86
2016-07-01	19197.938	1.7		1963-01-25	3.85
2016-10-01	19304.352	1.4		1963-01-28	3.86
2017-01-01	19398.343	1.9		1963-01-29	3.86
2017-04-01	19506.949	1.3		1963-01-30	3.86
2017-07-01	19660.766	1.2		1963-01-31	3.87
2017-10-01	19882.352	1.8		1963-02-01	3.88
2018-01-01	20044.077	2.6		1963-02-04	3.89
2018-04-01	20150.476	2.2		1963-02-05	3.90
2018-07-01	20276.154	1.3		1963-02-06	3.91
2018-10-01	20304.874	1.9		1963-02-07	3.91
2019-01-01	20415.150	1.6		1963-02-08	3.91
2019-04-01	20584.528	1.8		1963-02-11	3.92
2019-07-01	20817.581	1.5		1963-02-12	#N/A
2019-10-01	20951.088	1.3		1963-02-13	3.92
2020-01-01	20665.553	1.7		1963-02-14	3.91
2020-04-01	19034.830	-0.8		1963-02-15	3.91
2020-07-01	20511.785	3.1		1963-02-18	3.91
2020-10-01	20724.128	1.8		1963-02-19	3.93
2021-01-01	20990.541	3.6		1963-02-20	3.94
2021-04-01	21309.544	6.0		1963-02-21	3.95
2021-07-01	21483.083	4.8		1963-02-22	#N/A
2021-10-01	21847.602	5.2		1963-02-25	3.95
2022-01-01	21738.871	6.0		1963-02-26	3.95
2022-04-01	21708.160	4.7		1963-02-27	3.95
2022-07-01	21851.134	5.0		1963-02-28	3.94
2022-10-01	21989.981	4.7		1963-03-01	3.93
2023-01-01	22112.329	5.0		1963-03-04	3.92
2023-04-01	22225.350	3.7		1963-03-05	3.92
2023-07-01	22490.692	2.0		1963-03-06	3.91
2023-10-01	22668.986	2.1		1963-03-07	3.92