In lachlandeer/hayashir: R Companion to "Econometrics" by Fumio Hayashi

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

Application: Forward Exchange Rates as Optimal Predictors

Load the hayashir package

library(hayashir)

And other libraries we need for this chapter

library(dplyr) # data manipulation
library(ggplot2) # plotting
library(forecast) # time series plotting - Acf
library(sandwich) # HAC standard errors
library(lmtest)
library(car)

The Data

Let's get a quick look at our data by looking at the first 10 rows:

head(yen, 10)

Figure 6.1: Forecast Error: Yen/Dollar

ggplot(data = yen, aes(x = date, y = spot_30 - forward_30))+
    geom_line() +
    geom_hline(yintercept = 0, linetype="dashed", color = "red") +
    theme_bw()

Figure 6.2: Correlogram of $s30 - f$, Yen/Dollar

The Acf function from the forecast package will do what we need here:

Acf(yen$spot_30 - yen$forward_30, lag.max = 40)

Figure 6.3: Yen/Dollar Spot Rate, Jan 1975 - Dec 1989

ggplot(data = yen, aes(x = date, y = spot_rate))+
    geom_line() +
    scale_x_date(date_breaks = "3 years", date_labels =  "%m/%y") +
    theme_bw()

Figure 6.4: Plot of $s30$ against $f$, Yen/Dollar

ggplot(data = yen, aes(x = log(forward_30), y = log(spot_30)))+
    geom_point() +
    theme_bw()

Table 6.2: Regression Tests for Market Efficiency

For the Yen/Dollar:

mkt_eff <- lm (I(spot_30 - spot_rate) ~ I(forward_30 - spot_rate), data = yen)

summary(mkt_eff)

These results give standard errors under the assumption of heteroskedasticity. To correct for heteroskedasticity and autocorrelation we want HAC standard errors from the sandwich package. In particular standard errors with a maximum of 4 lags that are not pre-whitened. To get a summary of the regression we use the coeftest function from the lm package

coeftest(mkt_eff, vcov = vcovHAC(mkt_eff, lag = 4, prewhite = FALSE))

To test that $\beta_0 =0$ and $\beta_1 = 1$ we use the linearHypothesis function from car

linearHypothesis(mkt_eff, c("(Intercept) = 0", "I(forward_30 - spot_rate) = 1"),
                 vcov = vcovHAC(mkt_eff, lag = 4, prewhite = FALSE))

Figure 6.5: Plot of $s30-s$ against $f - s$, Yen/Dollar

ggplot(data = yen, aes(x = forward_30 - spot_rate, y = spot_30 - spot_rate))+
    geom_point() +
    geom_smooth(method = "lm", se = FALSE, color = "red") +
    theme_bw()