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<!DOCTYPE html>
<html lang="" xml:lang="">
<head>
<title>Forecasting-2-2---TV-Regression.utf8</title>
<meta charset="utf-8" />
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layout: true
.hheader[<a href="index.html"><svg style="height:0.8em;top:.04em;position:relative;fill:steelblue;" viewBox="0 0 576 512"><path d="M280.37 148.26L96 300.11V464a16 16 0 0 0 16 16l112.06-.29a16 16 0 0 0 15.92-16V368a16 16 0 0 1 16-16h64a16 16 0 0 1 16 16v95.64a16 16 0 0 0 16 16.05L464 480a16 16 0 0 0 16-16V300L295.67 148.26a12.19 12.19 0 0 0-15.3 0zM571.6 251.47L488 182.56V44.05a12 12 0 0 0-12-12h-56a12 12 0 0 0-12 12v72.61L318.47 43a48 48 0 0 0-61 0L4.34 251.47a12 12 0 0 0-1.6 16.9l25.5 31A12 12 0 0 0 45.15 301l235.22-193.74a12.19 12.19 0 0 1 15.3 0L530.9 301a12 12 0 0 0 16.9-1.6l25.5-31a12 12 0 0 0-1.7-16.93z"/></svg></a>]
---
class: center, middle, inverse
# Forecasting Time Series
## Time-varying Regression: Forecasting
.futnote[Eli Holmes, UW SAFS]
.citation[[email protected]]
---
Forecasting is easy in R once you have a fitted model.
Let's say for the anchovy, we fit the model
`$$C_t = \alpha + \beta t + e_t$$`
where `\(t\)` starts at 0 (so 1964 is `\(t=0\)` ). To predict, predict the catch in year t, we use
`$$C_t = \alpha + \beta t + e_t$$`
---
Model fit:
```r
model <- lm(log.metric.tons ~ t, data=anchovy)
coef(model)
```
```
## (Intercept) t
## 8.41962028 0.05818942
```
For anchovy, the estimated `\(\alpha\)` (Intercept) is 8.4196203 and `\(\beta\)` is 0.0581894. We want to use these estimates to forecast 1988 ( `\(t=24\)` ).
So the 1988 forecast is 8.4196203 + 0.0581894 `\(\times\)` 24 :
```r
coef(model)[1]+coef(model)[2]*24
```
```
## (Intercept)
## 9.816166
```
log metric tons.
---
# The forecast package
The forecast package in R makes it easy to create forecasts with fitted models and to plot (some of) those forecasts.
For a TV Regression model, our `forecast()` call looks like
```r
library(forecast)
fr <- forecast(model, newdata = data.frame(t=24:28))
```
---
The dark grey bands are the 80% prediction intervals and the light grey are the 95% prediction intervals.
```r
plot(fr)
```
<img src="Forecasting-2-2---TV-Regression_files/figure-html/plot.TVreg.forecast-1.png" style="display: block; margin: auto;" />
---
Sardine forecasts from a higher order polynomial can similarly be made. Let's fit a 4-th order polynomial.
`$$C_t = \alpha + \beta_1 t + \beta_2 t^2 + \beta_3 t^3 + \beta_4 t^4 + e_t$$`
To forecast with this model, we fit the model to estimate the `\(\beta\)`'s and then replace `\(t\)` with `\(24\)`:
`$$C_{1988} = \alpha + \beta_1 24 + \beta_2 24^2 + \beta_3 24^3 + \beta_4 24^4 + e_t$$`
---
This is how to do that in R:
```r
model <- lm(log.metric.tons ~ t + I(t^2) + I(t^3) + I(t^4), data=anchovy)
fr <- forecast(model, newdata = data.frame(t=24:28))
fr
```
```
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1 10.18017 9.856576 10.50377 9.670058 10.69028
## 2 10.30288 9.849849 10.75591 9.588723 11.01704
## 3 10.41391 9.770926 11.05689 9.400315 11.42750
## 4 10.50839 9.609866 11.40691 9.091963 11.92482
## 5 10.58101 9.354533 11.80748 8.647599 12.51442
```
---
Unfortunately, forecast does not recognize that there is only one predictor `\(t\)` and we cannot use forecast's plot function.
If you do this in R, it throws an error.
```r
try(plot(fr))
```
```
## Error in plotlmforecast(x, PI = PI, shaded = shaded, shadecols = shadecols, :
## Forecast plot for regression models only available for a single predictor
```
```
Error in plotlmforecast(x, PI = PI, shaded = shaded, shadecols = shadecols, : Forecast plot for regression models only available for a single predictor
```
---
I created a function that you can use to plot time-varying regressions with polynomial `\(t\)`. You will use this function in the lab.
```r
plotforecasttv(model, ylims=c(8,17))
```
<img src="Forecasting-2-2---TV-Regression_files/figure-html/plot.TVreg.forecast2-1.png" style="display: block; margin: auto;" />
---
A feature of a time-varying regression with many polynomials is that it fits the data well, but the forecast quickly becomes uncertain due to uncertainty regarding the polynomial fit. A simpler model can give forecasts that do not become rapidly uncertain.
The flip-side is that the simpler model may not capture the short-term trends very well and may suffer from autocorrelated residuals.
```r
model <- lm(log.metric.tons ~ t + I(t^2), data=sardine)
```
---
```r
plotforecasttv(model, ylims=c(8,17))
```
<img src="Forecasting-2-2---TV-Regression_files/figure-html/plot.TVreg.lm1-1.png" style="display: block; margin: auto;" />
---
# Summary
* Time-varying regression is a simple approach to forecasting that allows a non-linear trend.
* The uncertainty in your forecast is determined by how much error there is between the fit an the data.
* Fit must be balanced against prediction uncertainty.
* R allows you to quickly fit models and compute the prediction intervals.
Careful thought must be given to selecting the polynomial order.
* Standard methods are available in R for order selection
* Using different orders for different data sets has prediction consequences
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