See attached PDF and Excel datasetHON-1...

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See attached PDF and Excel datasetHON-1 	10/21/19	1252.26001	1254.629028	1240.599976	1246.150024	1246.150024	1027200 	10/22/19	1247.849976	1250.599976	1241.380005	1242.800049	1242.800049	1023800 	10/23/19	1242.359985	1259.890015	1242.359985	1259.130005	1259.130005	911500 	10/24/19	1260.900024	1264	1253.714966	1260.98999	1260.98999	1028100 	10/25/19	1251.030029	1269.599976	1250.01001	1265.130005	1265.130005	1213100 	10/28/19	1275.449951	1299.310059	1272.540039	1290	1290	2613200 	10/29/19	1276.22998	1281.589966	1257.212036	1262.619995	1262.619995	1886400 	10/30/19	1252.969971	1269.359985	1252	1261.290039	1261.290039	1408900 	10/31/19	1261.280029	1267.670044	1250.843018	1260.109985	1260.109985	1455700 	11/1/19	1265	1274.619995	1260.5	1273.73999	1273.73999	1670100 	11/4/19	1276.449951	1294.130005	1276.35498	1291.369995	1291.369995	1501000 	11/5/19	1292.890015	1298.930054	1291.229004	1292.030029	1292.030029	1282700 	11/6/19	1289.459961	1293.72998	1282.5	1291.800049	1291.800049	1152700 	11/7/19	1294.280029	1323.73999	1294.244995	1308.859985	1308.859985	2030000 	11/8/19	1305.280029	1318	1304.36499	1311.369995	1311.369995	1251400 	11/11/19	1303.180054	1306.425049	1297.410034	1299.189941	1299.189941	1011900 	11/12/19	1300	1310	1295.77002	1298.800049	1298.800049	1085900 	11/13/19	1294.069946	1304.300049	1293.51001	1298	1298	826700 	11/14/19	1297.5	1317	1295.650024	1311.459961	1311.459961	1193500 	11/15/19	1318.939941	1334.880005	1314.280029	1334.869995	1334.869995	1782600 	11/18/19	1332.219971	1335.529053	1317.5	1320.699951	1320.699951	1487400 	11/19/19	1327.699951	1327.699951	1312.800049	1315.459961	1315.459961	1269200 	11/20/19	1311.73999	1315	1291.150024	1303.050049	1303.050049	1308600 	11/21/19	1301.47998	1312.589966	1293	1301.349976	1301.349976	995500 	11/22/19	1305.619995	1308.72998	1291.410034	1295.339966	1295.339966	1385700 	11/25/19	1299.180054	1311.310059	1298.130005	1306.689941	1306.689941	1036200 	11/26/19	1309.859985	1314.800049	1305.089966	1313.550049	1313.550049	1069700 	11/27/19	1315	1318.359985	1309.630005	1312.98999	1312.98999	995600 	11/29/19	1307.119995	1310.204956	1303.969971	1304.959961	1304.959961	587000 	12/2/19	1301	1305.829956	1281	1289.920044	1289.920044	1510900 	12/3/19	1279.569946	1298.46106	1279	1295.280029	1295.280029	1143800 	12/4/19	1307.01001	1325.800049	1304.869995	1320.540039	1320.540039	1537500 	12/5/19	1328	1329.358032	1316.439941	1328.130005	1328.130005	1212700 	12/6/19	1333.439941	1344	1333.439941	1340.619995	1340.619995	1314800 	12/9/19	1338.040039	1359.449951	1337.839966	1343.560059	1343.560059	1354300 	12/10/19	1341.5	1349.974976	1336.040039	1344.660034	1344.660034	1094100 	12/11/19	1350.839966	1351.199951	1342.670044	1345.02002	1345.02002	850400 	12/12/19	1345.939941	1355.775024	1340.5	1350.27002	1350.27002	1281000 	12/13/19	1347.949951	1353.093018	1343.869995	1347.829956	1347.829956	1549600 	12/16/19	1356.5	1364.680054	1352.670044	1361.170044	1361.170044	1397300 	12/17/19	1362.890015	1365	1351.322998	1355.119995	1355.119995	1854000 	12/18/19	1356.599976	1360.469971	1351	1352.619995	1352.619995	1522600 	12/19/19	1351.819946	1358.099976	1348.984985	1356.040039	1356.040039	1469900 	12/20/19	1363.349976	1363.640015	1349	1349.589966	1349.589966	3315000 	12/23/19	1355.869995	1359.800049	1346.51001	1348.839966	1348.839966	883100 	12/24/19	1348.5	1350.26001	1342.780029	1343.560059	1343.560059	347500 	12/26/19	1346.170044	1361.327026	1344.469971	1360.400024	1360.400024	667500 	12/27/19	1362.98999	1364.530029	1349.310059	1351.890015	1351.890015	1038400 	12/30/19	1350	1353	1334.02002	1336.140015	1336.140015	1050900 	12/31/19	1330.109985	1338	1329.084961	1337.02002	1337.02002	961800 	1/2/20	1341.550049	1368.140015	1341.550049	1367.369995	1367.369995	1406600 	1/3/20	1347.859985	1372.5	1345.543945	1360.660034	1360.660034	1186400 	1/6/20	1350	1396.5	1350	1394.209961	1394.209961	1732300 	1/7/20	1397.939941	1402.98999	1390.380005	1393.339966	1393.339966	1502700 	1/8/20	1392.079956	1411.579956	1390.839966	1404.319946	1404.319946	1528000 	1/9/20	1420.569946	1427.329956	1410.27002	1419.829956	1419.829956	1500900 	1/10/20	1427.560059	1434.928955	1418.349976	1429.72998	1429.72998	1820700 	1/13/20	1436.130005	1440.52002	1426.02002	1439.22998	1439.22998	1652300 	1/14/20	1439.01001	1441.800049	1428.369995	1430.880005	1430.880005	1558900 	1/15/20	1430.209961	1441.39502	1430.209961	1439.199951	1439.199951	1282700 	1/16/20	1447.439941	1451.98999	1440.920044	1451.699951	1451.699951	1173700 	1/17/20	1462.910034	1481.295044	1458.219971	1480.390015	1480.390015	2233600   Microsoft Word - Document10 Deliverables:  R Studio script       Word.docx as described with the plots and narrative outlined  below  Applications of Quadratic Programming The Hodrick-Prescott Filter (Decomposition) is a  mathematical method used in real business cycle theory in economics to decompose a time  series into its cyclical and trend components. Its formulation is based on the following quadratic  programming problem: Let be the logarithm of a time series (hence, itself a time series). 1. Perform research about the Hodrick-Prescott decomposition and provide  insights about its historical development in Word docx.   2. Interpret each term of the Hodrick-Prescott objective function, and discuss a  few advantages and disadvantages of this decomposition method in Word.docx.   3. Consider the quarterly time series of Alphabet (GOOG) stock prices (courtesy  of https://finance.yahoo.com given in the Excel workbook: Project-part 2-  Data.xlsx. Apply the Hodrick-Prescott optimization method to decompose the  logarithm of the given time series into its cyclic and trend components. Use R  Studio to solve the problem.   4. Interpret the results obtained from step 3, above, and discuss the merits of  your decomposition, in a Word docx.

Pooja · Accepted Answer

1)
Hodrick-Prescott filter removes the cyclical component in a time series data. It gives a smoothed-curve. Hodrick-Prescott filter was introduced in the year 1923 by E. T. Whittaker. However, its major use was in the year 1990’s.
2)
Yt = Tt + Ct + Et
Where, Yt is log of a time series variable
Tt = trend component 
Ct = cyclic component
Et = error component
Hodrick-Prescott filter objective is to minimize 
Such that  = penalty for cyclic component.
= penalty for variations in the growth rate of the trend component. 
Lambda = 100*(number of periods in a year)2
lambda = 100*(3652) = 13322500 for a daily time series
The advantage of Hodrick-Prescott filter are
· removes the cyclic component
The disadvantages of Hodrick-Prescott filter are 
· Data still contains I(2) trend
· Noise follows normal distribution
· Predictions are misleading [as algorithm changes while minimizing in iterations] 
3)
> library("readxl")
> apple  apple$lnprice  
> library(mFilter)
> library(quantmod)
> hp  hp
Title:
 Hodrick-Prescott Filter 
Call:
 hpfilter(x = apple$lnprice, freq = 13322500)
Method:
 hpfilter
Filter Type:
 lambda
Series:

HON-1 10/21/19 XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX1027200 10/22/19 XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX1023800 10/23/19 XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX...

Answer To: HON-1 10/21/19 XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX1027200 10/22/19 XXXXXXXXXX...

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