## 5.9 Exponential Smoothing

NOTE: I don’t know where this section is supposed to go.

Suppose we have an ongoing time series $$y_1, y_2, \dots$$ and we want to make a prediction about $$y_{t+1}$$ given the past $$y_t, y_{t-1}, y_{t-2}, \dots$$. The exponential smoothing approach takes a value $$\lambda\in (0, 1)$$ and produces

$\hat{y}_{t+1} = (1-\lambda)\sum_{j=0}^\infty \lambda^j y_{t-j}$ This forecasted value weights the previous values with the weights decreasing geometricallly as the values go further back in time.

Although the sum is infinite, we can see readily that

$\begin{eqnarray*} \hat{y}_{t+1} & = & (1-\lambda) \left[ y_t + \lambda y_{t-1} + \lambda^2 y_{t-2} + \cdots \right]\\ & = & (1-\lambda)y_t + (1-\lambda)\sum_{j=1}^\infty\lambda^jy_{t-j}\\ & = & (1-\lambda)y_t + \lambda(1-\lambda)\sum_{j=0}^\infty\lambda^jy_{t-1-j}\\ & = & (1-\lambda)y_t + \lambda \hat{y}_t \end{eqnarray*}$

So from the infinite sum we can derive a rather simple recursion. Furthermore, this recursion can be re-written as

$\begin{eqnarray*} \hat{y}_{t+1} & = & (1-\lambda)y_t + \lambda \hat{y}_t - \hat{y}_t + \hat{y}_t\\ & = & (1-\lambda)y_t -\hat{y}_t (1-\lambda) + \hat{y}_t\\ & = & \hat{y}_t + (1-\lambda)(y_t-\hat{y}_t) \end{eqnarray*}$

Here, the prediction for time $$t+1$$ is written as the predicted value for time $$t$$ plus the deviation of the observed $$y_t$$ and the predicted $$\hat{y}_t$$. In this way, the exponential smoother is incorporating new data while giving some weight to the predicted value. It is clear that if $$\lambda=1$$, we stick with our predicted values and if $$\lambda=0$$ we go with the new observed value. For values between $$0$$ and $$1$$ we average between the two.

One way to choose $$\lambda$$ is to choose it based on how far back you want values to influence the current prediction. For example, for a “90-day” exponential smooth, we might choose $$\lambda$$ such that $$\lambda^j < \varepsilon$$ for $$j > 90$$ and $$\varepsilon$$ very small. For example, for $$\varepsilon = 0.00001$$, we would want $$\lambda < 0.00001^{1/90} \approx 0.88$$.

A common stock chart for “technical analysis” is to plot a stock’s observed prices along with various exponential smooths with differing weights. If the current observed price is above one of the smoothed averages it might be considered “over-bought”, while if the observed price is below one of the averages it might be considered “over-sold”. Be forewarned that this is not financial advice!

Here is a plot of the weighted mid price for the SPY exchange traded fund for a few seconds of trading.

Here is the same plot with two exponential smooths, one with $$\lambda=0.995$$ and one with $$\lambda=0.999$$.