HW2

1. Estimate …

#dm1

dm1 %>% 
  transmute(
    Y = log(S30/S), 
    X = log(`F`/S)
  ) -> 
  god1

god1 %>% 
  lm(Y ~ X, data = .) -> 
  lm_god1

lm_god1 %>% 
  summary()

## 
## Call:
## lm(formula = Y ~ X, data = .)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.114915 -0.019740  0.002125  0.019888  0.100061 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.011315   0.002472  -4.578 5.47e-06 ***
## X           -3.014681   0.662965  -4.547 6.30e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.03343 on 776 degrees of freedom
## Multiple R-squared:  0.02595,    Adjusted R-squared:  0.0247 
## F-statistic: 20.68 on 1 and 776 DF,  p-value: 6.303e-06

lm_god1$residuals %>% 
  ts() %>% 
  as.timeSeries() -> 
  ts_god1

ts_god1$SS.1 %>% 
  acf(main="ACF of index returns")

Ans. Yes, the autocorrelation appear to vanish after 4 lags.

2. Please test …

test whether \(\beta_{0} = 0\)

lm_god1 %>% 
  coeftest(vcov. = vcovHAC, type = "HC1")

## 
## t test of coefficients:
## 
##               Estimate Std. Error t value Pr(>|t|)  
## (Intercept) -0.0113149  0.0047611 -2.3765  0.01772 *
## X           -3.0146811  1.2492680 -2.4132  0.01605 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

test whether \(\beta_{1} = 1\)

library(multcomp)

glht(
  lm_god1, linfct = "X = 1"
) %>% 
  summary()

## 
##   Simultaneous Tests for General Linear Hypotheses
## 
## Fit: lm(formula = Y ~ X, data = .)
## 
## Linear Hypotheses:
##        Estimate Std. Error t value Pr(>|t|)    
## X == 1   -3.015      0.663  -6.056 2.18e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## (Adjusted p values reported -- single-step method)