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xaringan_presentation.Rmd
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---
output:
xaringan::moon_reader:
lib_dir: libs
nature:
highlightStyle: github
highlightLines: true
---
```{r setup, include=FALSE, message=FALSE}
options(htmltools.dir.version = FALSE, servr.daemon = TRUE)
```
class: center, middle, inverse
# Forecasting Time Series
## Time-varying Regression
.futnote[Eli Holmes, UW SAFS]
.citation[[email protected]]
---
```{r load_data, echo=FALSE}
load("landings.RData")
landings$log.metric.tons = log(landings$metric.tons)
landings = subset(landings, Year <= 1989)
```
## Time-varying regression
Time-varying regression is simply a linear regression where time is the explanatory variable:
$$log(catch) = \alpha + \beta t + \beta_2 t^2 + \dots + e_t$$
The error term ( $e_t$ ) was treated as an independent Normal error ( $\sim N(0, \sigma)$ ) in Stergiou and Christou (1996). If that is not a reasonable assumption, then it is simple to fit an autocorrelated error model or non-Gausian error model in R.
---
Stergiou and Christou (1996) fit time-varying regressions to the 1964-1987 data and show the results in Table 4.
---
The first step is to determine how many polynomials of $t$ to include in your model.
```{r poly.plot, echo=FALSE,fig.height=4,fig.width=8,fig.align="center"}
par(mfrow=c(1,4))
tt=seq(-5,5,.01)
plot(tt,type="l",ylab="",xlab="")
title("1st order")
plot(tt^2,type="l",ylab="",xlab="")
title("2nd order")
plot(tt^3-3*tt^2-tt+3,ylim=c(-100,50),type="l",ylab="",xlab="")
title("3rd order")
plot(tt^4+2*tt^3-12*tt^2-2*tt+6,ylim=c(-100,100),type="l",ylab="",xlab="")
title("4th order")
```
---
Here is how to fit a linear regression to the anchovy landings with a 4th-order polynomial for time. We are fitting this model:
$$log(Anchovy) = \alpha + \beta t + \beta_2 t^2 + \beta_3 t^3 + \beta_4 t^4 + e_t$$
```{r tvreg.anchovy}
landings$t = landings$Year-1963
model <- lm(log.metric.tons ~ poly(t,4),
data=landings, subset=Species=="Anchovy"&Year<=1987)
```
---
They do not say how they choose the polynomial order to include. We will look at the fit and keep the significant polynomials.
```{r}
summary(model)
```
---
This suggests that we keep only the 1st polynomial, i.e. a linear relationship with time.
```{r tvreg.anchovy2}
dat = subset(landings, Species=="Anchovy" & Year <= 1987)
model <- lm(log.metric.tons ~ t, data=dat)
```
The coefficients and adjusted R2 are similar to that shown in Table 4.
```{r}
c(coef(model), summary(model)$adj.r.squared)
```
---
We want to test if our residuals are independent. We can do this with the Ljung-Box test as Stergio and Christou (1995) do. Stergio and Christou appear to use a lag of 14 for the test (this is a bit large for 24 data points). The degrees of freedom is lag minus the number of estimated parameters in the model. So for the Anchovy data, $df = 14 - 2$.
```{r}
x <- resid(model)
Box.test(x, lag = 14, type = "Ljung-Box",fitdf=2)
```
Compare to the values in the far right column in Table 4.
---
For the sardine (bottom row in Table 4), Stergio and Christou fit a 4th order polynomial. There are two approaches you can take to fitting n-order polynomials. The first is to use the `poly()` function. This creates orthogonal covariates for your polynomial.
What does that mean? Let's say you want to fit a model with a 2nd order polynomial of $t$. It has $t$ and $t^2$, but using these are highly correlated. They also have different means and different variances, which makes it hard to compare the estimated effect sizes. The `poly()` function creates covariates with mean and covariance or zero and identical variances.
```{r poly}
T1 = 1:24; T2=T1^2
c(mean(T1),mean(T2),cov(T1, T2))
T1 = poly(T1,2)[,1]; T2=poly(T1,2)[,2]
c(mean(T1),mean(T2),cov(T1, T2))
```
---
With `poly()`, a 4th order time-varying regression model is fit to the sardine data as:
```{r tvreg.sardine}
dat = subset(landings, Species=="Sardine" & Year <= 1987)
model <- lm(log.metric.tons ~ poly(t,4), data=dat)
```
This indicates support for the 2nd, 3rd, and 4th orders but not the 1st (linear) part.
---
```{r poly.summary}
summary(model)
```
---
However, Stergiou and Christou used a raw polynomial model using $t$, $t^2$, $t^3$ and $t^4$ as the covariates. We can fit this model as:
```{r tvreg.sardine2}
dat = subset(landings, Species=="Sardine" & Year <= 1987)
model <- lm(log.metric.tons ~ t + I(t^2) + I(t^3) + I(t^4), data=dat)
```
The coefficients and adjusted R2 are similar to that shown in Table 4.
```{r}
c(coef(model), summary(model)$adj.r.squared)
```
---
The test for autocorrelation of the residuals is
```{r}
x <- resid(model)
Box.test(x, lag = 14, type = "Ljung-Box",fitdf=5)
```
`fitdf` specifies the number of parameters estimated by the model. In this case it is 5, intercept and 4 coefficients.
The p-value is less than 0.05 indicating lack of autocorrelation in the residuals.
---
class: center, middle, inverse
# Summary
---
## Why use time-varying regression?
* It looks there is a simple time relationship. If a high-order polynomial is required, that is a bad sign.
* Easy and fast
* Easy to explain
* You are only forecasting a few years ahead
* No assumptions required about 'stationarity'
---
## Why not to use time-varying regression?
* Autocorrelation is not modeled. That autocorrelation may hold information for forecasting.
* You are only using temporal trend for forecasting (mean level).
* If you use a high-order polynomial, you might be modeling noise from a random walk. That means interpreting the temporal pattern as having information when in fact it has none.
## Is time-varying regression used?
All the time. Most "trend" analyses are a variant of time-varying regression. If you fit a line to your data and report the trend or percent change, that's a time-varying regression.