Evaluation of ARMA Prediction Errors

We compare the "next-day" ARMA prediction results and a simple Random Walk (RW) model. In RW model the next day value is $y_{t+1}=y_t+\epsilon_{t+1}$ and the predicted value is equal to the conditional expectation of $y_{t+1}=y_t$ . This means that the present value is the RW prediction for the next day.

There are two reasons to consider RW models

• simplicity of RW
• RW is a trivial case of ARMA where $q=0$, $p=1$, and $a_1=1$.
• RW represents unpredictable time series, because random numbers $\epsilon_{t+1}$ are independent.

Therefore the difference of average prediction errors of ARMA and RW models can be regarded as some parameter of "ARMA-unpredictability" of the data. That means that if this difference is zero or negative then the time series are not predictable by the ARMA model.

Table 1.1: The average "next-day" prediction results of ARMA and RW models

$Data$	$DeltaARMA \%$	$MeanARMA$	$VarARMA$
$/$\pounds)$	- 1.779090e-01	1.293609e+00	8.454827e-02
DM/$	- 1.191e-02	1.092e-01	9.985e-02
Yen/$	- 1.086e+00	6.369e+00	6.446505e+00
Fr/$	- 3.029e-01	4.285e-01	3.395e-01
AT&T	-1.375e+00	4.554e+00	3.621e+00
Intel Co	+2.814e-01	2.052e+01	1.936e+00
Hermis Bank	-4.280e+01	2.374e+01	1.998e+01
London Stock Exchange	-5.107e-01	2.751e+02	2.346e+01
Call Center	+3.076e+01	8.453e+02	7.111e+02

Table 1.1 shows the difference between the mean square deviations of ARMA and RW models using

• exchange rates of DM/$, $/$\pounds$, Yen/$, and Fr/$
• closing rates of AT&T , Intel Co., and Hermis Bank stocks
• index of the London Stock Exchange
• call rates of a call center.

In Table 1.1 the symbol $MeanARMA$ denotes average prediction errors of ARMA model. The symbol $VarArma$ means the variance of ARMA predictions. The symbol $DeltaARMA \%$ denotes the relation $(RW-ARMA)/RW\%$ in percentages, where $RW$ defines average errors of the Random Walk. If ARMA predicts better then $DeltaARMA\%>0$.

The data was divided into three equal parts.
The first part was applied to estimate parameters $a$ and $b$ of an ARMA model using different parameters $p$ and $q$.
The best values of $p$ and $q$ were defined using the second part of data.
The third part was used to compare ARMA and RW models. The table shows the comparison results.

Table 1.1 demonstrates that the ARMA model predicts all the financial data not better than RW. However ARMA predicts call rates 31 percent better then RW. That is a statistically significant difference. The observed deviations between RW and ARMA models predicting financial data are to small for practical conclusions.

Figure 1.1: The optimal parameters of the ARMA model predicting call rates.

Figure 1.1 shows optimal parameters $b$ and $a$. In this case the optimal $q=1$ and $p=12$.

The "multi-day" predictions of call rates are shown in Figure 1.2.

Figure 1.2: Multi-day predictions of call rates