Since
mid-March, markets have continued to recover. Following the CARES act passed on
March 27, 2020 by the US Federal Government, investments in stocks increased
rapidly. Now, before the controversial second stimulus plan hits, index prices
seem to remain dormant. Owing to the course of the oncoming elections, market
pricing behavior could change, and investors may want to consider proceeding
with caution once they decide to invest in the stock market.
The usual perception of market pricing on stocks suggests that if the current president is elected, prices will spike, but if the opposing candidate is elected, prices will sharply decline with only a slight increase overtime. For the purpose of this article, however, the methods used have not been predicated on the outcome of the November 3rd election, rather, I have attempted to conduct a general analysis to forecast the results of the S&P 500 price index by applying the Box-Jenkins method.
An answer to the question “how long will we wait for the market to reach the 4000s?” can be shown by the forecasted price index demonstrated in this article. ARMA(11,23) and EGARCH(11,23) models both predict that there will be new record highs in prices by April 2021, similarly, both models reflect potential record high prices in the 4000s by February 2022 on average. These forecasts are independent of election results and different presidential scenarios. Let us dive into forecasting along with the Box-Jenkins methodology by starting with a unit root test to determine the level of integration for the S&P 500 price index series.
For this analysis, I use the ADF unit root test, which is one of the most reliable tests to define unit root with a time series, for the index; it shows that if a time series has a unit root, it is not stationary or vice versa. Since the S&P 500 price index series has a unit root, it can be inferred that the market is not stationary at level. Hence, the first difference of the price index series should be examined to understand whether it is stationary. If so, we may continue practicing fairly well technical forecasts using a stationary time series.
According to the results of the ADF unit root test, the price index series does not have a unit root at the first difference, so the S&P 500 price index becomes a stationary time series after this formulation. Stationary time series are used in econometrics for making reasonable forecasts. This rule is particularly important for ARMA and ARCH models.
The Box-Jenkins method assumes that if a time series is stationary, we may continue building proper models to make convenient forecasts. The S&P 500 price index series for the period 2008M1-2020M10 is stationary at first difference. Therefore, forecasts are made of this stationary series which means that the difference of the natural logarithm of the original price index series is run for the entire analysis.
The second step of the method is to define which ARMA terms should be selected for a solid model so that we could be able to make coherent forecasts. Taking a look at the correlogram of the series helps us to find out which ARMA terms should be used for generating an outstanding model.
CORRELOGRAM
OF S&P PRICE INDEX SERIES
The correlogram found in table 1 indicates significant spikes for some lags such as 6th lag, 15th lag, and 22nd lag. These lagged values are preferred to be in autoregressive moving average models and autoregressive conditional variance models. I tried various combinations to find the most appropriate model based on the smallest Akaike information criteria. Hence, I ended up making a forecast with ARMA(11,23) for the index.
The third step of the Box-Jenkins method is to check the diagnosis of residuals to see if the residuals of the models are normally distributed and homoscedastic and whether they have no serial correlation. Since financial time series practice leptokurtic distribution in which kurtosis is larger than that of a normal distribution, it is not deemed as a matter for the models. Yet, forecasts made with ARMA and ARCH models are perceived as arts more than science. Both models do not practice autocorrelation and heteroscedasticity, and they are stable. The last step of the method requires forecast analysis.
Graph 1 shows the forecasted returns of the S&P 500 index for the period 2020M11-2022M10. As you can see in Graph 1, the time series indicating returns of the S&P 500 index has almost zero mean, therefore it is stationary and can be technically analyzed. The ARMA(11,23) model expects negative returns at the end of November, December 2020, and January 2021, and then the index will see positive returns again way through 2021 with some fluctuations in May and July.
Graph 2 shows the prices of the S&P 500 index created by forecasted returns for the period 2020M11-2022M10. The price will not hit the last peak of the index until April 2021 in terms of the ARMA(11,23) model’s forecast. However, recovery resumes as of March 2021 and the index sees its record high of 3923 by the end of January 2022. Prices move steadily during 2022 with some fluctuations in March and April. 4053 is tested by October 2022.