Question

(1) Explain smoothing methods in time series. How to find the opti- mal value of the smoothing parameter in an exponentially weighted moving average. (2) Prove that ARCH(1) process is leptokurtic. Also establish its equiv- alence to AR process. (3) Prove that exponentially weighted moving average model is a special case of GARCH(1,1) processes. Also establish its equivalence to ARMA(1,1) process. (4) Find the minimum mean square error (MMSE) forecast of AR(2) process and then obtain the variance of the forecast error.

Answer 1

Answer: (1)

• Smoothing method in time series:

Smoothing method in time series is the technique to remove the fine-grained variation between time steps. By this method we smooth out the irregular roughness ( remove noise ) to obtain clearer signal.

The most common smoothing used generally in time series analysis is ' Moving averages '. This smoothing technique is also used in time series forecasting.

Sometimes the term ' filter ' is also used to describe a smoothing procedure.

• The optimal value of smothing parameter in an Exponentially Weighted Moving Average ( EWMA ):

The result of Exponentially Weighted Moving Average is cumulative as it contains the previously calculated average and therefore all the data points contributes to the result.

In EWMA each squared return is weighted by a multiplier. This multiplier is called the smoothing parameter in EWMA. Usually it is denoted by lamda $(\lambda )$ .

$\lambda$ shows the rate at which the older data will come into the overall calculation. The value of $\lambda$ must be less than 1 and typically $\lambda$ is in between 0 to 1.

If $\lambda = 1$ that means only the most recent data has been used to measure EWMA.

If $\lambda \rightarrow 0$   that means more weightage is given to older data

If $\lambda \rightarrow 1$ that means newer data has been given more weightage.

If a day wise data range is given , then the smoothing parameter

$\lambda =\frac{2}{Sum \; of\; the\; length\; of\; the\; days}$

Typically for a 7 day range,   $\lambda =\frac{2}{1+7}= 0.25$

The EWMA Formula to calculate value of moving average at a time t is

$EWMA(t)= \lambda x(t)+(1-\lambda )EWMA(t-1)$

or, more generally

$Current \; period\; EWMA = \lambda ( Current\; period \; data\; value ) +(1-\lambda )\; ( previous\; period\; EWMA)$