Forecasting-2-1-TV-Regression.Rmd

# 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 a non-Gausian error model in R.

#### Order of the time polynomial {-}

The order of the polynomial of $t$ determines how the time axis (x-axis) relates to the overall trend in the data.  A 1st order polynomial ($\beta t$) will allow a linear relationship only.  A 2nd order polynomial($\beta_1 t + \beta_2 t^2$) will allow a convex or concave relationship with one peak.  3rd and 4th orders will allow more flexible relationships with more peaks.

```{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")
```