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I am curious in which cases (if any) you would force an exponential smoothing function to only incorporate data from the past year? Per Holt method, one would continue to use the full time series as opposed to a 1-year lookback.

Is the idea behind this that the "trend" component would update the forecast for any structural shifts? If my data has a totally different monthly trend this past year as opposed to prior years, wouldn't I want to restrict Holt Winters method to only observe months from the past year?

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Depending on what tool you are using for Holt you can always set the value for $\beta$ close to $0.9$ to make it more reactive to recent changes.

Keep in mind the formula for Holt's method:

$$\hat{Y}_{t+1} = \alpha Y_{t} + (1-\alpha)(\hat{L}_{t}+\hat{T}_{t})$$ $$\hat{T}_{t} = \beta (Y_{t}-Y_{t-1})+(1-\beta)\hat{T}_{t-1}$$

$Y_t$ is the actual value, $\hat{Y}_{t}$ is the estimated/forecast value, $\hat{L}_{t}$ is the estimated level, and $\hat{T}_{t}$ is the estimated trend. $\beta$ is the trend smoothing factor, $0 < \beta <1$.

$\beta$ close to $0$ tends to give a long term average of the trend, while $\beta$ close to $1$ gives you a value that is almost a copy of the most recent value, with little consideration of previous values. Hence a high $\beta$ means the trend is more reactive to recent changes.

Most modern tools will find $\beta$ automatically using an optimization routine, but you could simply manually set it to $0.9$ or something.

This answers your basic question, but doesn't really solve your underlying challenge which is detecting underlying structural changes.

Making the trend very reactive might help in capturing structural shifts, but there are better ways to do so with change point detection and structural break detection methods.

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