Hypothesis Testing: Exercises
1. Hypothesis testing. Assume you have performed a linear regression and saved the
output to the object ols_results, which has been defined for you in this exercise.
You now want to summarize the regression output using the summary() function.
You do not need to write any code, but check that you are able to complete the following tasks:
- Use the t-stat or p-value to evaluate whether the slope coefficient is significant at the 5% level.
- Use the t-stat or p-value to evaluate whether the y-intercept is significant at the 1% level.
date <- c("Jan-02","Feb-02","Mar-02","Apr-02","May-02","Jun-02","Jul-02","Aug-02","Sep-02","Oct-02",
"Nov-02","Dec-02","Jan-03","Feb-03","Mar-03","Apr-03","May-03","Jun-03","Jul-03","Aug-03","Sep-03",
"Oct-03","Nov-03","Dec-03","Jan-04","Feb-04","Mar-04","Apr-04","May-04","Jun-04","Jul-04","Aug-04",
"Sep-04","Oct-04","Nov-04","Dec-04","Jan-05","Feb-05","Mar-05","Apr-05","May-05","Jun-05","Jul-05",
"Aug-05","Sep-05","Oct-05","Nov-05","Dec-05","Jan-06","Feb-06","Mar-06","Apr-06","May-06","Jun-06",
"Jul-06","Aug-06","Sep-06","Oct-06","Nov-06","Dec-06","Jan-07","Feb-07","Mar-07","Apr-07","May-07",
"Jun-07","Jul-07","Aug-07","Sep-07","Oct-07","Nov-07","Dec-07","Jan-08","Feb-08","Mar-08","Apr-08",
"May-08","Jun-08","Jul-08","Aug-08","Sep-08","Oct-08","Nov-08","Dec-08","Jan-09","Feb-09","Mar-09",
"Apr-09","May-09","Jun-09","Jul-09","Aug-09","Sep-09","Oct-09","Nov-09","Dec-09","Jan-10","Feb-10",
"Mar-10","Apr-10","May-10","Jun-10","Jul-10","Aug-10","Sep-10","Oct-10","Nov-10","Dec-10","Jan-11",
"Feb-11","Mar-11","Apr-11","May-11","Jun-11","Jul-11","Aug-11","Sep-11","Oct-11","Nov-11","Dec-11",
"Jan-12","Feb-12","Mar-12","Apr-12","May-12","Jun-12","Jul-12","Aug-12","Sep-12","Oct-12","Nov-12",
"Dec-12","Jan-13","Feb-13","Mar-13","Apr-13","May-13","Jun-13","Jul-13","Aug-13","Sep-13","Oct-13",
"Nov-13","Dec-13","Jan-14","Feb-14","Mar-14","Apr-14","May-14","Jun-14","Jul-14","Aug-14","Sep-14",
"Oct-14","Nov-14","Dec-14","Jan-15","Feb-15","Mar-15","Apr-15","May-15","Jun-15","Jul-15","Aug-15",
"Sep-15","Oct-15","Nov-15","Dec-15","Jan-16","Feb-16","Mar-16","Apr-16","May-16","Jun-16","Jul-16",
"Aug-16","Sep-16","Oct-16","Nov-16","Dec-16","Jan-17","Feb-17","Mar-17","Apr-17","May-17","Jun-17",
"Jul-17","Aug-17","Sep-17","Oct-17","Nov-17","Dec-17","Jan-18","Feb-18")
sp500 <- c(1130.199951,1106.72998,1147.390015,1076.920044,1067.140015,989.820007,
911.619995,916.070007,815.280029,885.76001,936.309998,879.820007,855.700012,
841.150024,848.179993,916.919983,963.590027,974.5,990.309998,1008.01001,995.969971,
1050.709961,1058.199951,1111.920044,1131.130005,1144.939941,1126.209961,1107.300049,
1120.680054,1140.839966,1101.719971,1104.23999,1114.579956,1130.199951,1173.780029,
1211.920044,1181.27002,1203.599976,1180.589966,1156.849976,1191.5,1191.329956,
1234.180054,1220.329956,1228.810059,1207.01001,1249.47998,1248.290039,1280.079956,
1280.660034,1302.880005,1310.609985,1270.050049,1270.060059,1278.530029,1303.800049,
1335.819946,1377.76001,1400.630005,1418.030029,1437.900024,1406.800049,1420.829956,
1482.369995,1530.619995,1504.660034,1455.180054,1473.959961,1527.290039,1545.790039,
1479.630005,1467.969971,1378.599976,1330.449951,1326.410034,1385.969971,1399.619995,
1276.689941,1269.420044,1287.829956,1164.170044,968.669983,888.609985,902.98999,
823.090027,729.570007,793.590027,872.73999,923.26001,920.820007,990.219971,1019.52002,
1054.910034,1036.180054,1098.890015,1116.560059,1073.890015,1105.359985,1171.22998,
1188.579956,1087.300049,1031.099976,1107.530029,1049.719971,1143.48999,1185.709961,
1186.599976,1257.619995,1289.140015,1328.640015,1329.47998,1365.209961,1345.199951,
1320.640015,1292.589966,1219.119995,1131.209961,1251,1246.910034,1258.859985,1312.449951,
1365.900024,1408.469971,1397.859985,1309.869995,1362.329956,1379.319946,1406.540039,
1440.900024,1412.199951,1416.339966,1426.189941,1498.109985,1514.680054,1569.180054,
1597.550049,1631.709961,1609.780029,1689.420044,1635.949951,1682.410034,1758.699951,
1806.550049,1845.859985,1782.680054,1857.680054,1873.959961,1884.390015,1923.869995,
1962.290039,1929.800049,2004.069946,1971.439941,2018.209961,2065.780029,2058.899902,
1996.670044,2105.22998,2067.629883,2087.379883,2108.639893,2067,2104.48999,1970.089966,
1919.650024,2080.76001,2082.929932,2038.199951,1936.939941,1937.089966,2056.620117,
2067.169922,2093.939941,2099.340088,2173.149902,2171.330078,2164.330078,2128.679932,
2200.169922,2251.570068,2285.590088,2380.129883,2362.340088,2388.5,2415.649902,
2431.389893,2477.100098,2474.419922,2521.199951,2583.209961,2645.100098,2683.72998,
2816.449951,2715.219971)
microsoft <- c(31.855,29.17,30.155001,26.129999,25.455,27.35,23.99,24.540001,21.870001,26.735001,
28.84,25.85,23.73,23.700001,24.209999,25.57,24.610001,25.639999,26.41,26.52,27.799999,
26.139999,25.709999,27.370001,27.65,26.530001,24.93,26.129999,26.23,28.559999,
28.49,27.299999,27.65,27.969999,26.809999,26.719999,26.280001,25.16,24.17,25.299999,
25.799999,24.84,25.610001,27.379999,25.73,25.700001,27.68,26.15,28.15,26.870001,
27.209999,24.15,22.65,23.299999,24.059999,25.700001,27.35,28.709999,29.360001,29.860001,
30.860001,28.17,27.870001,29.940001,30.690001,29.469999,28.99,28.73,29.459999,
36.810001,33.599998,35.599998,32.599998,27.200001,28.379999,28.52,28.32,27.51,25.719999,
27.290001,26.690001,22.33,20.219999,19.440001,17.1,16.15,18.370001,20.26,20.889999,
23.77,23.52,24.65,25.719999,27.73,29.41,30.48,28.18,28.67,29.290001,30.540001,
25.799999,23.01,25.809999,23.469999,24.49,26.67,25.26,27.91,27.73,26.58,25.389999,
25.92,25.01,26,27.4,26.6,24.889999,26.629999,25.58,25.959999,29.530001,31.74,32.259998,
32.02,29.190001,30.59,29.469999,30.82,29.76,28.540001,26.620001,26.709999,27.450001,
27.799999,28.610001,33.099998,34.900002,34.540001,31.84,33.400002,33.279999,35.41,38.130001,
37.41,37.84,38.310001,40.990002,40.400002,40.939999,41.700001,43.16,45.43,46.360001,
46.950001,47.810001,46.450001,40.400002,43.849998,40.66,48.639999,46.860001,44.150002,
46.700001,43.52,44.259998,52.639999,54.349998,55.48,55.09,50.880001,55.23,49.869999,
53,51.169998,56.68,57.459999,57.599998,59.919998,60.259998,62.139999,64.650002,63.98,
65.860001,68.459999,69.839996,68.93,72.699997,74.769997,74.489998,83.18,84.169998,
85.540001,95.010002,93.769997)
microsoft = 100*diff(log(microsoft))
sp500 = 100*diff(log(sp500))
N = length(date)
log_returns = data.frame(date[2:N], sp500, microsoft)
log_rf <- 100*log(1.0+0.5/100.0)
log_returns$y <- log_returns$microsoft - log_rf
log_returns$x <- log_returns$sp500 - log_rf
ols_results <- lm(y ~ x, data = log_returns)
# Summarize regression results.
summary(___)
# Summarize regression results.
summary(ols_results)
test_error()
success_msg("Good job!")
summary() takes the output of the regression as its input.Previous Exercise Next Exercise