Econometric models
If, in January 1926, an investor had placed $1 in one-month U.S. Treasuries — one of the "safest" assets in the world — and continued to reinvest the proceeds in Treasury bills month after month until December 2001, the $1 investment would have increased to $14. If, on the other hand, the investor had invested $1 in the stock market, for example, the S&P 500, and continued to reinvest the proceeds in the stock market month by month over the same seventy-year period, the $1 investment would have grown to $1,918, a considerably larger sum.
Now suppose that each month, an investor could say in advance which of these two investments would produce a higher return for that month, and leverages that information by changing the current total of their initial $1 investment in the higher-yielding asset, month by month.
What would be the return on a $1 investment in such a "perfect" investment strategy by December 2023?
The surprising answer – 55,823,861,439 (yes, more than $55 billion for a $1 bet; that's not a typographical error) – is often a shock to even the most experienced professional investment manager.
Of course, no investor has perfect foresight. But this extreme example suggests that even a modest ability to predict returns on financial assets can be amply rewarded.
For this reason, quantitative models of the risks and rewards of financial investments – now known collectively as financial technology or financial engineering – have become virtually indispensable for institutional investors around the world. We want to extend the scope of these instruments to the retail investor. Econometric theory and models can be considered as essential tools for understanding and trying to predict stock price movements.
The advantage of these techniques over other modes of stock selection is that they make it possible to understand changes in share prices in a cold, coherent and reasoned way. This is a guarantee of success in the medium or long term. We applied these techniques to different asset classes: S&p500, Nasdaq, gold and gold mining stocks. The purpose of these procedures is to obtain an equation that can explain the behavior of an asset (S&P500, mining stocks, etc.) called an independent variable as a combination of dependent variables (interest rate, offset values of S&p500, etc.).
Asset price = b0+b1V1+b2V2+...+BnVn+ residuals
Observations are collected on a monthly, quarterly and annual basis. The field of analysis is vast ranging from 1966 to 2023 for the monthly and quarterly models. The annual model runs from 1905 to 2023. As a result, a very large number of bull and bear markets are taken into account, which gives credence to the patterns. We use the NCSS statistical processing software to manufacture our models.
These mathematical models are proprietary, but it is nevertheless possible to present 7 principles that govern them.
Principle 1 Hendry/ERP approach and multi-collinearity
We start with a general model with many variables and then rduce the model by eliminating the explanatory variable with the least significant coefficient (T Value). This approach, from general to simple, has been called the Hendry/ERP approach. We start with a general dynamic model that is over-parameterized, then we perform a simplification via Wald and t tests. This allows us to discard insignificant variables in order to improve the precision of other variables and reduce the multicollinearity that may be present. There was evidence of multicollinearity in the regression of many of our models. Nevertheless, over the entire sampling period, the correlation between the independent factors is not excessive.
The signs of the explanatory variables are mostly in line with expectations. Therefore, although multicollinearity is a research problem and may influence some of the results, it does not determine the majority of the results. The effect of multi-collinearity between the explanatory variables is to increase the standard errors of the regression coefficients and to reduce the t-statistics, thus making the coefficients less significant (and perhaps even insignificant) of the variables to be included in the equation. One way to reduce its impact is to reason on real values, deflated by rising prices.
Principle 2 Preference for models using Weighted Regression techniques
The Ordinary Least Squares (OLS) method is popular because of its ease of use. This method can thus be used in Excel. However, traditional regression methods do not adequately account for the occurrence of outlier values (values that fall outside the range of most of the data). Outliers frequently occur in real-world data and can go unnoticed without careful inspection or filtering. The regression coefficients of the equation can then be significantly affected by the presence of outliers, and OLS methods offer no resistance to extreme values.
If the hypothesis of homoscedasticity is rejected (a common occurrence regarding financial time series), we are faced with the problem of trying to find alternative estimation procedures superior to ordinary least squares (available in Excel). Heteroscedasticity, that is, a non-constant variance or alternating periods of low and high volatility, has been found for all models (S&P 500, Nasdaq, Gold). We therefore decided to use weighted least squares (WLS Method) to obtain consistent and asymptotically more efficient estimates of the parameters of the equation B0, B1, B21 ... Bn as described in the introduction.
Principle 3 Measuring Gaps
Our models do not claim to predict the exact future price changes of an asset. What our models provide is the equilibrium price of an asset (S&P500, Nasdaq, etc.) month after month, quarter after quarter, or year after year. It is interesting to measure the difference between the real and expected value of the asset because this difference tells us about the speculative component in price formation. It also has predictive power that we can rely on once certain thresholds (confidence intervals in statistical terms) have been crossed. At a certain point, the price increase tends to turn into a speculative bubble which is an unsustainable increase in price caused by investors' buying behavior rather than by real fundamental information captured by the explanatory variables.
Regarding the S&P500 index, which sets the tone at the global level, including for the French CAC of course, the strongest values of the indices of the monthly and annual models were:
monthly annual
model model
11 December 1972 1.06 1.30 1973-1974 crash
10 November 1980 1.14 1.25
25 August 1987 1.14 1.30 1987 crash
March 24, 2000 1.14 1.35 Internet Crash
11 Oct 2007 1.07 1.18 Subprime crash
February 19, 2020 1.04 1.21
November 5, 2021 1.08 1.2
The lows corresponding to the upward reversal point of the market have been:
monthly annual
model model
15 December 1974 0.83 0.70
12 August 1982 0.96 0.76
15 November 1987 0.78 0.72
October 9, 2002 0.83 0.66 End of the Internet Crash
November 21, 2008 0.69 0.46 Outlier
March 23, 2020 0.74 0.71
*** Gold mine valuation models yield:
Downturn
August 19, 1974 1.22 1.60 Outlier
10 October 1980 1.37 1.31
17 January 1983 1.29 1.22
3 August 1987 1.25 1.30
13 Jan 1994 1.18 1.28
4 June 2002 1.11 1.74
February 2, 2006 1.16 1.27
9 November 2010 1.20 1.18
August 12, 2016 1.23 1.21
August 5, 2020 1.36 1.29
Upward reversal
24 August 1976 0.71 0.65
21 June 1982 0.71 0.58
18 Jan 1993 0.92 0.88
9 December 1997 0.82 0.85
October 28, 2008 0.54 0.58 Outlier
September 7, 2018 0.90 0.76
September 26, 2022 0.88 0.77
We recommend the calculation of an average between the monthly index and the annual index in order to assess the over- or undervaluation of stock market indices (S&P500, gold mines).
Principle 4 Transfer model
Time series analysis and multiple regression techniques can be combined to produce a better prediction than would be possible with the use of either technique alone. The idea is to build an Arima model for the residuals of the regression equation. This dual model is likely to provide better forecasts since it includes both a structural explanation (based on observable economic variables) of the formation of an asset's price (gold mines, S&p500) and a temporal explanation of the part of the variance (residue analysis) that cannot be explained by known macroeconomic factors. The residues are analysed and projected using statistical procedures called "ARIMA".
Principle 5 Rationale for the 9-month growth variable
The models take into account the phenomenon of amplification that results from speculation based on recent capital gains, which is particularly important for gold mines. There is a process of self-amplification in the stock market that is superimposed on traditional economic factors. So, once economic factors have initiated a price increase: "Speculation will then determine other new purchases. Initial buyers, eager to make their speculative gains, will be easily relayed by new layers of buyers. At each stage, the previous increase will serve as the basis and the beginning of a new increase. This is an essential phenomenon (Vigneaux). "This process shows that the rise or fall in prices feeds on itself. But this process cannot be indefinite. There is a force of recall. Indeed, individuals become aware of the exaggerations committed, which gradually leads them to correct their valuations, thus triggering a market reversal. A cumulative downward process then begins to develop. A similar but opposite phenomenon occurred at the end of major bear markets (December 1974, August 1982, November 1987, October 2002, March 2003, March 2020 for the S&P500 or the Nasdaq, similar points for the CAC40 given the preponderance of Wall Street at the global level).
Principle 6 Lagged Variables
The problem of positive autocorrelation has been encountered in all the models we have designed. It is caused by the fact that random shocks that are not fully explained by the independent variables (or explanatory variables) tend to have a persistent effect on the dependent variable (S&p500 for example). The fact that a positive error remains positive in the following months can be attributed to inertia or resistance to change on the part of investors. We also notice that the adaptation of a linear relationship to a relationship that often conforms to a nonlinear relationship leads to the presence of autocorrelation and the need to address this problem. Unlike multi-collinearity, which places a downward bias on measured T's, positive autocorrelation results in upward-biased measured T's. The use of lagged variables is also justified by the use of charts and technical analysis by many traders in the market – technical analysis is the practice of predicting future price movements by trying to spot geometric shapes and other regularities in historical price charts.
Finally, it can be assumed that many economic or financial phenomena are influenced by anchoring. The phenomenon of anchoring would seem relevant to the theory of "sticky prices" that macroeconomists talk about
The anchoring phenomenon would seem relevant to the theory of "sticky prices" that macroeconomists talk about. As long as past prices are considered suggestions for new prices, new prices will tend to be close to past prices. The more ambiguous the value of a product, the more important a suggestion is likely to be, and the more important anchoring is likely to be for price determination.
Principle 7 Use of dummy variables
The relationship between dependent and independent variables can undergo a structural change also known as structural instability or structural break, i.e., the relationship changes from one period to the next. With respect to the gold mining model, the regulation of the price of gold prior to the Nixon decision in 1971 resulted in a significant reduction in the value of gold stocks from 1966 to 1972. The coefficient of the dummy variable (1 from 1966 to 1971 for the monthly and quarterly gold stock models, 0 otherwise) was both negative and significant at the 1% level. Another structural break in November 1980 (the peak of the US inflation rate) taken into account with a discrete variable (value 0 or 1) was also observed for the S&P500 valuation model.
Principle 8 Use of AI
We use CHATGPT 5.4, enhanced by the ADA module (a gpts module), Advanced Data Analysis, to analyze and predict the evolution of the residuals of our structural models (see the notion of transfer model - principle 4).
The link below provides an example of processing by AI (ARIMA Modeling). Click Here
We are attaching here a study describing the remarkable performance of AI models (ChatGPT 4, for example) in linear and nonlinear regression. We will comment on this article soon on the blog of this site.
