Double moving average smoothing time series

Question

I´m reading: Regression modeling with actuarial and financial applications by Frees, and in page 274 he talks about a double smoothing procedure for a times series with linear trend:

Suppose that we have a time series data with linear trends of the form: $$y_t = \beta_0 + \beta_1 t +\epsilon_t$$

We can use a double smoothing moving average procedure to forecast this time series:

1. Create a smoothed time series with the original observations, that is: $$s^1_t = (y_t + · · · + y_{t−k+1})/k$$

2. Create a doubly smoothed series by treating the smoothed series created in step (1) as input. That is:

$$s^2_t = (s^1_t + · · · + s^1_{t−k+1})/k$$

This procedure smooths out the effect of a linear trend in time.The estimate of the trend is

$$b_{1,T} = \frac{2(s^1_T-s^2_T)}{k-1}$$

and the resulting forecasts are:

$$y_{T+l} = s^1_T + b_{1,T} l$$ for forecasts l lead time units into the future.

My questions are: How do we arrive to this formula of double smoothing procedure? The books does not contain any proof about this fact, it just puts the algorithm and that's it. Where do this algorithm comes from? How do we know that this procedure smooths out the effect of a linear trend in time?

I would really appreciate any hints or suggestions with this problem.

Answer 1

I'm reading Frees for Exam SRM and I found this part similarly confusing. You can show this with some arithmetic sequence formulas.

Start with $s_t = \dfrac{y_t + y_{t-1} + \cdots + y_{t-k-1}}{k}$.

Because we're assuming $y_t$ comes from a linear trend in time model, $y_t = \beta_0 +\beta_1 t, y_{t-1} = \beta_0 +\beta_1 (t-1)$, etc. (I'm ignoring the $\epsilon_t$ terms because I figure they get wiped out by the smoothing procedure). Use arithmetic sequence formulas to add up all the $y_t$. Don't forget to divide by $k$ in the denominator of the smoothing formula. I get $s_t = \beta_0 + \beta_1 (t - \frac k2 + \frac 12)$.

Next, let's work on $s_t^2 = \dfrac{s_t + s_{t-1} + \dots + s_{t-k-1}}{k}$. We just found that $s_t = \beta_0 + \beta_1 (t - \frac k2 + \frac 12)$, so $S_{t-1} = \beta_0 + \beta_1 ((t-1) - \frac k2 + \frac 12)$, etc. Use arithmetic sequence formulas to add up all the $s_t^2$. Don't forget to divide by $k$ in the denominator of the smoothing formula. I get $s_t^2 = \beta_0 + \beta_1 t + \beta_1 - k \beta_1$.

Now we can check Frees' formula. $s_t - s_t^2$ is $\frac k2 \beta_1 - \frac 12 \beta_1$ or $\dfrac{(k-1)\beta_1}{2}$. So it really is true that $\dfrac{2(s_t - s_t^2)}{k-1}$ is an estimate of $\beta_1$.

Answer 2

If you start with a linear trend in time model:

$Y_{t}=\beta_{0}+\beta_{1}t+\epsilon_{t} \rightarrow y_{t}=\beta_{0}+\beta_{1}t$

and use the equations for a single/double smoothing arithmetic average:

$\hat{s}_{n}(k) = \dfrac{1}{k} \sum_{i=1}^{i=k} y_{n-i+1} = \dfrac{1}{k} \sum_{i=1}^{i=k} \beta_{0}+\beta_{1} \cdot (n-i+1)$

$=\beta_{0}+\beta_{1} (n-(\frac{k-1}{2}))$

and

$\hat{s}^{(2)}_{n}(k) = \dfrac{1}{k^2} \sum_{j=n-k+1}^{j=n}\Big[ \sum_{i=1}^{i=k}y_{j-i+1}\Big] = \dfrac{1}{k^2} \sum_{j=n-k+1}^{j=n}\Big[ \sum_{i=1}^{i=k}\beta_{0}+\beta_{1} \cdot (j-i+1)\Big]$

$=\beta_{0}+\beta_{1} (n-(k-1))$

The estimate of the trend is given by:

$b_{1}=\dfrac{\hat{s}_{n}(k)-\hat{s}^{(2)}_{n}(k)}{(k-1)/2}=\dfrac{\Big( \beta_{0}+\beta_{1} (n-(\frac{k-1}{2}))\Big)-\Big( \beta_{0}+\beta_{1} (n-(k-1))\Big)}{(k-1)/2}=\beta_{1}$

But the estimate for the intercept (level) should be:

$b_{0}=2\hat{s}_{n}(k)-\hat{s}^{(2)}_{n}(k)=2\Big( \beta_{0}+\beta_{1} (n-(\frac{k-1}{2}))\Big)-\Big( \beta_{0}+\beta_{1} (n-(k-1))\Big)=\beta_{0}+\beta_{1}n=y_{n}$

and not:

$b_{0}=\hat{s}_{n}(k)=\beta_{0}+(n-\frac{k-1}{2})\beta_{1} \ne y_{n}$

as this will correctly make the forecast:

$\hat{y}_{n+l} = b_{0}+b_{1} \cdot l = \beta_{0} +\beta_{1} \cdot n + \beta_{1} \cdot l = \beta_{0} +\beta_{1} \cdot (n+l) = y_{n+l}$

Notice this form of $b_{0}$ is the same as the one given by the double exponential smoothing procedure.

I have contacted Dr. Frees about this correction but he is simply referencing (pg 288):

Abraham, Bovas, and Ledolter, Johannes (1983). Statistical Methods for Forecasting. John Wiley & Sons, New York.

I attempted to look up the derivation for $b_{0}$ in the 1st edition of Statistical Methods for Forecasting, but none is given (only double exponential smoothing estimates can be found).