1.2 Import the data

Gas_prices <- readxl::read_xlsx(path = "data/Gas_prices_1.xlsx")
head(Gas_prices)

tail(Gas_prices)

if (!require("forecast")){install.packages("forecast")}

L1_Crude_price <- Gas_prices$Crude_price[-1]
Unleaded <- Gas_prices$Unleaded[-dim(Gas_prices)[1]]
ts_Unleaded <- ts(Unleaded, start=1996, frequency = 12)

1.3 Time-series exploration

plot(ts_Unleaded) # Nonstationary time-series

seasonplot(ts_Unleaded) # Strong evidence of seasonality

1.4 Scatter plot between Unleaded and L1_Crude_price

plot(Unleaded~L1_Crude_price)

1.5 Simple Linear Model

model1 <- lm(Unleaded ~ L1_Crude_price) # obtain least square estimate
plot(L1_Crude_price,Unleaded,pch=20) 
abline(model1, col="red")

1.6 Interpretation of Least Square Estimates

summary(model1)

## 
## Call:
## lm(formula = Unleaded ~ L1_Crude_price)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -78.109 -14.510  -5.058  11.189  90.272 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    67.17064    3.43101   19.58   <2e-16 ***
## L1_Crude_price  2.84481    0.05435   52.34   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 26.13 on 237 degrees of freedom
## Multiple R-squared:  0.9204, Adjusted R-squared:   0.92 
## F-statistic:  2740 on 1 and 237 DF,  p-value: < 2.2e-16

1.7 Fitted Values and Graphical Display

Merge1 <- cbind(Unleaded, Fitted=model1$fitted.values)
rownames(Merge1) <- Gas_prices$Date[-dim(Gas_prices)[1]]
head(Merge1)

##             Unleaded   Fitted
## 15-JAN-1996    109.0 121.4780
## 15-FEB-1996    108.9 127.8504
## 15-MAR-1996    113.7 134.0236
## 15-APR-1996    123.1 127.3952
## 15-MAY-1996    127.9 125.2616
## 15-JUN-1996    125.6 127.7650

plot(L1_Crude_price, Unleaded, pch=20, ylim = c(90,450)) 
points(L1_Crude_price, model1$fitted.values, pch=4)
abline(model1, col="red")

1.8 Interpretation of Fitted Values

Merge2 <- cbind(Merge1, Residuals=model1$residuals)
head(Merge2)

##             Unleaded   Fitted  Residuals
## 15-JAN-1996    109.0 121.4780 -12.478008
## 15-FEB-1996    108.9 127.8504 -18.950376
## 15-MAR-1996    113.7 134.0236 -20.323607
## 15-APR-1996    123.1 127.3952  -4.295207
## 15-MAY-1996    127.9 125.2616   2.638398
## 15-JUN-1996    125.6 127.7650  -2.165032

plot(L1_Crude_price, Unleaded, pch=20, ylim = c(90,450))
points(L1_Crude_price, model1$fitted.values, pch=4)
for (i in 1:dim(Gas_prices)[1])
{
  lines(c(L1_Crude_price[i], L1_Crude_price[i]),
        c(model1$fitted.values[i],Unleaded[i]), col="red")
}

1.9 Coefficient of Determination, \(R^2\)

For the gas price data set, we have built a simple linear regression between the gasoline price y and lagged crude oil price. However, how strong is the relationship between x and y? To answer this question, we need coefficient of determination, \(R^2\).

\[y_i-\bar{y} = (y_i - \hat{y}_i) + (\hat{y}_i -\bar{y})\]

\[\underbrace{\sum_{i=1}^{n} (y_i-\bar{y})^2}_{SST} = \underbrace{\sum_{i=1}^{n} (y_i - \hat{y}_i)^2}_{SSE} + \underbrace{\sum_{i=1}^{n} (\hat{y}_i -\bar{y})^2}_{SSR}\]

Here is a figure illurstrating this decomposition.

Figure: R squre explanation

1.10 Definition of \(R^2\)

\(R^2\) is defined as

\[R^2=\frac{SSR}{SST} = \frac{\sum_{i=1}^{n} (\hat{y}_i -\bar{y})^2}{\sum_{i=1}^{n} (y_i-\bar{y})^2} = 1-\frac{SSE}{SST}\]