Time-series Forecasting and Predictive Modelling

Problem Statement

The objective of this project is to analyse the performance of the U.S. stock and bond returns over the period from December 1979 to December 2021, and to generate out-of-sample excess return forecasts based on different predictive variables and models. The analysis includes computing various statistics for the stock and bond returns, generating time-series of monthly out-of-sample constant expected excess return forecasts using a recursive estimate approach and rolling window estimate approach, and using five plausible predictors to generate monthly out-of-sample excess return forecasts for each asset class.

The five plausible predictors used in the analysis of equity market are the inflation (infl), book-to-market ratio (BM), stock variance (SVAR), net equity expansion (NTIS), and dividend-payout ratio (D/E). Additionally, for the bond market, the analysis includes the variables of inflation (infl), Treasury bill rate (TBL), long-term yield (LTY), long-term return (LTR) and default yield spread (DFY).

The importance of analysing the performance of the U.S. stock and bond returns lies in their impact on the economy and financial markets. Investors use stock and bond returns as a measure of their in- vestment performance, while policy-makers use them as a gauge of the overall health of the economy. Additionally, the project’s analysis of the predictive models and variables can provide insights into the factors that drive stock and bond returns, helping investors make informed investment decisions.

Data Sources

monthly prices of the S&P 500 stock market index (SP500) (hereafter referred as the stock index) from the U.S. Federal Reserve Economic Data (FRED) website.

monthly prices of the Bloomberg Barclays U.S. Aggregate Bond Index (LBUSTRUU) (hereafter referred as the bond index) from Bloomberg.

monthly data on the risk-free rate of return from Professor Kenneth French’s data library at Dartmouth College.

The variables used in the analysis include annualised mean/average, annualised volatility, annualised Sharpe ratio, skewness, and kurtosis.The analysis is carried out in two parts- in Part A - Recursive Estimate Approach is used and in Part B – Rolling Window Estimate Approach is adopted.

Summary Statistics of data

Summary Statistic U.S. stock and bond simple excess returns

Over the period from December 1979 to December 2021, the stock index provided higher mean excess returns (6.23% per annum) compared to the bond index (3.22% per annum).

However, the stock index was also more volatile (15.08%) than the bond index (5.11%), which resulted in a lower Sharpe ratio (0.41) compared to the bond index (0.63).

The stock index excess returns experienced a negative skewness while the bond index excess returns were positively skewed over the same period.

The kurtosis of the excess returns was high for both indices, indicating that the distribution of returns was fat-tailed and had more extreme values.

1. Recursive Estimation Approach

Recursive estimation involves training a model on a historical period from y(0) to y(n) and predicting the value of yˆ(n + 1). The model is then retrained using the extended historical period from y(0) to y(n + 1) to predict yˆ(n + 2), and this process is repeated until the end of the sample period. The window of historical data used for training the model increases with each prediction, and only one-step-ahead predictions are made.

The total sample was divided into two periods: an in-sample period from January 1980-December 1999 and an out-of-sample period from January 2000-December 2021. A time-series of monthly out-of-sample constant expected excess return forecasts (mean benchmark forecasts) were generated using the recursive estimation approach as described above for both the assets.

1. 1. Modelling:

The corresponding five plausible predictors were used to generate monthly out-of-sample excess return forecasts for each of the asset classes. In total, eight different regression models were trained for each asset class to compare and contrast their respective predictive performances.

OLS predictive regression model for each of the five predictors - 5 models

Combination forecasts of excess returns that is a simple average of the forecasts based on the five predictors from the OLS model

Penalised linear regression models - 2 models

LASSO model - performs variable selection by shrinking the regression coefficients of some predictors to zero. It is particularly useful when the number of predictors is large, and some of them may not be relevant in explaining the variation in the response variable.
Ridge model - adds a penalty term to the least squares estimation, which is proportional to the square of the magnitude of the coefficients. The penalty term helps to prevent overfitting by shrinking the coefficients towards zero but not necessarily to zero. Ridge regression is often used when the predictors are highly correlated, and there is a risk of multicollinearity.

Recursive Predictive Models for SP500 (Stock index)

Recursive Predictive Models for LBUSTRUU (Bond index)

The analysis of predictive models for both stock and bond indices revealed limited statistical significance and explanatory power for the examined variables, indicating their limited impact on return variations.

1. 2. Model Evaluation:

For both asset classes, the benchmark model and the other eight recursive predictive models’ MSFE (Mean Squared Forecast Error) values are computed. The average forecast error for each model is measured by the MSFE values, which may be used to assess how well the models estimate the returns on the assets. Also the ratio of MSFE’s of predictive models to benchmark MSFE values is calculated. This allows for a comparison of model accuracy, with a ratio >1 indicating poorer performance and <1 indicating better performance.

Ratios of MSFE for predictive models relative to benchmark for SP500 and LBUSTRUU

The Diebold-Mariano (DM) test was utilised to assess the equal predictive ability of all eight recursive predictive models compared to the mean benchmark forecasts for both the asset classes. The null hypothesis for the Diebold-Mariano (DM) test is that there is no difference in the predictive accuracy of two models, while the alternative hypothesis is that one model has superior predictive accuracy compared to the other. In other words, the DM test is used to determine whether the difference in forecast accuracy between two models is statistically significant, with the null hypothesis assuming that there is no significant difference, and the alternative hypothesis assuming that there is a significant difference.

Upon reviewing p-values for all the models, it appears that most of the models don’t exhibit statistically significant differences in forecast accuracy compared to the benchmark for both the asset classes. The p-values are relatively high, suggesting that the differences observed in the mean squared forecast error (MSFE) values are likely due to random variations rather than meaningful distinctions in predictive ability.

For the stock index, high volatility (dip in returns) is noted around 2008 crisis for all the model forecasts, but rest of the return series appear fairly stable. But on the contrary, volatility has been noted throughout the return series for the bond index.

1. 3. Portfolio and Asset Allocation:

A portfolio of the two assets is formed, represented by their corresponding excess returns for all the different model forecasts. The recursive method is used to generate out-of-sample covariance matrix forecasts for the portfolio. The covariance matrix forecast provides valuable information about the volatility and the relationship between the returns of the portfolio assets. These insights are crucial for portfolio managers and investors as they assist in asset allocation decisions and risk management.

The monthly out-of-sample excess return forecasts and corresponding portfolio asset weights for all predictive model forecasts is generated in a recursive manner. The weights are computed in the context of an Optimal Tangency Portfolio for all the models respectively.

Summary statistics of monthly out-of-sample excess return forecasts for the Optimal Tangency Portfolio

The ratio of asset allocation weights in the two asset portfolio under the LASSO model indicates higher allocation to stocks from year 2000 and rapidly decreasing to become equally weighted around the year 2003. From there on, the allocation to bonds has been higher than stocks. With the other models, however, the asset allocation strategy has been to overweight bonds over stocks.

The cumulative excess return of the portfolio has been below the benchmark, albeit increasing over the years. The portfolio excess return closely followed the benchmark until the year 2008 for all the predictive models, but has underperformed the benchmark thereafter, nonetheless trending upwards over time.

2. Rolling Window Estimation Approach

Rolling estimation, on the other hand, involves training a model on a fixed window size, albeit a sliding window of historical data. It first trains on data from y(0) to y(n) and predicts the value of yˆ(n + 1). The model is then retrained using the next window of historical data from y(1) to y(n + 1), ensuring that window size remains the same, to predict yˆ(n + 2), and this process is repeated until the end of the sample period. The size of the historical window used for training the model remains fixed, and only one-step-ahead predictions are made.

2. 1. Modelling:

The same eight predictive models are used to train on the in-sample data, albeit they are trained using a rolling-window approach as opposed to a recursive method. The window size is fixed to the initial in-sample data size, which is 240 monthly observations in our case.

Rolling Predictive Models for SP500 (stock index)

Rolling Predictive Models for LBUSTRUU (bond index)

The results for the rolling window predictive models indicate that most of the predictors are not statistically significant in explaining the returns of either of the indices. Only a few predictors show moderate significance (Model 5 for the bond index), but their explanatory power is limited. Overall, the models have low to very limited explanatory power in predicting the returns of both asset classes.

2. 2. Model Evaluation:

Based on the MSFE values and the DM test results, most of the predictive models, do not exhibit statistically significant differences in performance based on the p-values. The MSFE ratios indicate only slight variations in performance compared to the benchmark.

The monthly excess returns for the stock index forecasted using rolling window approach projected larger negative returns (loss) during the 2008 financial crisis relative to the recursive estimation forecasts.

2. 3. Portfolio and Asset Allocation:

The ratio of asset allocation weights in the two asset portfolio for the LASSO model forecasts, using rolling window estimation, shows a similar pattern/strategy to the recursive approach, however, allocation to bonds was higher than for stocks. The allocation under the combination and benchmark models mirror each other.

The magnitude of the upward movement varies across models with LASSO indicating lower returns over time. The returns have been more dispersed for the rolling window estimation approach relative to the recursive approach.

Conclusion

In conclusion, the asset allocation strategy for the two asset portfolio, based on LASSO model forecasts, shifts from overweighting stocks to bonds around the early 2000s. The portfolio’s cumulative excess return has been below the benchmark but increasing over time. The performance of the portfolio varies across models, with the LASSO model indicating lower returns. Further refinement of the models and asset allocation strategies is needed to improve portfolio performance and potentially outperform the benchmark.

The python code for the above implementation is open source and can be easily accessed using this link.