Time-Invariance Coefficients Tests with the
Adaptive Multi-Factor Model^††thanks: We thank Dr. Manny Dong, and all the supports from Cornell University. Declarations of interest: none.

We use the ETFs to select the basis assets to be included in the multi-factor model. For comparison to the literature and as a robustness test, we include the Fama-French 5 factors into this set⁵⁵5For the FF5 factors, we add back the risk-free rate if it is already subtracted in the raw data as in the market return in Fama-French’s website https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html..

The data used in this paper consists of all the stocks and all the ETFs available in the CRSP (The Center for Research in Security Prices) database over the years 2007 - 2018. We start in 2007 because prior the number of ETFs with a sufficient amount of capital (in other word, tradable) is limited, which makes it difficult to fit the AMF model.

To avoid the influence of market microstructure effects, we use a weekly observation interval. A security is included in our sample only if it has prices and returns available for more than 2/3 of all the trading weeks. This is consistent with the empirical asset pricing literature. Our final sample consists of over 4000 companies listed on the NYSE. To avoid a survivorship bias, we include the delisted returns (see Shumway (1997) [18] for more explanation). For each regression time period, we form the adjusted price using Equation (2).

We repeat our analysis on all sub-periods of 2007 - 2018 (both ends inclusive, the same below) starting at Jan. 1st, ending at Dec. 31th, and with a length at least 3 years. This means we have 55 sub-periods in total, such as 2007 - 2009, 2008 - 2010, …, 2008 - 2018, 2007 - 2018. We only use the time periods with length at least 3 years to ensure that there are sufficient observations to fit the models. As stated above, we use the weekly observations to avoid the market microstructure effects, so 3 years means $n\geq 156$ observations in such regression.

The number of ETFs in our sample is large, slightly over 2000 by the end of 2018, i.e. the number of basis assets $p_{1}>2000$. The set of potential factors is denoted $V_{j}(1\leq j\leq p_{1})$. Since $n=156<2000<p_{1}$ for the 3-year time periods, high-dimensional statistical methods, including the GIBS algorithm, need to be used.

This section provides a brief review of the high-dimensional statistical methodologies used in this paper, including the GIBS algorithm.

Let $\left\|\bm{v}\right\|_{d}$ denote the standard $l_{d}$ norm of a vector, then

$$\left\|\bm{v}\right\|_{d}=\left(\sum_{i}|\bm{v}_{i}|^{d}\right)^{1/d}\qquad\text{where}\;\;0<d<\infty,$$

(7)

specifically, $\left\|\bm{v}\right\|_{1}=\sum_{i}|\bm{v}_{i}|$. Suppose $\bm{\beta}$ is a vector with dimension $p\times 1$. Given a set $S\subseteq\{1,2,...,p\}$, we let $\beta_{S}$ denote the $p\times 1$ vector with i-th element

$$(\bm{\beta}_{S})_{i}=\begin{cases}\beta_{i},&\text{if}\;\;i\in S\\ 0,&otherwise.\end{cases}$$

(8)

Here the index set $S$ is called the support of $\bm{\beta}$, in other words, $supp(\bm{\beta})=\{i:\bm{\beta}_{i}\neq 0\}$. Similarly, if $\bm{X}$ is a matrix instead of a vector, then $\bm{X}_{S}$ are the columns of $\bm{X}$ indexed by $S$. Denote the ones vector $\mathbbm{1}_{n}$ as a $n\times 1$ vector with all elements being 1, $\bm{J_{n}}=\mathbbm{1}_{n}\mathbbm{1}_{n}^{\prime}$, and $\bm{\bar{J}_{n}}=\frac{1}{n}\bm{J_{n}}$. $\bm{I_{n}}$ denotes the identity matrix with diagonal 1 and 0 elsewhere. The subscript $n$ is always omitted when the dimension $n$ is clear from the context. The notation $\#S$ means the number of elements in the set $S$.

For any matrix $M=\{m_{i,j}\}_{i,j=1}^{n}$ define the following terms

	k-th skew-diagonal	$\displaystyle=\{m_{i,j}|i=j+k\}$		(9)
	k-th skew-anti-diagonal	$\displaystyle=\{m_{i,j}|i=n+k+1-j\}$		(10)

where $-n+1\leq k\leq n-1$.

Because of this high-dimension problem and the high-correlation among the basis assets, traditional methods fail to give an interpretable and systematic way to fit the AMF model. Therefore, we employ the GIBS algorithm proposed in the paper Zhu et al. (2020) [24] to select the basis assets set for each stock .

We give a brief review of the GIBS algorithm in this section. In the GIBS algorithm, a procedure using the Minimax-Linkage Prototype Clustering is employed to obtain low-correlated ETFs (denoted as $U$ in the paper). The high-dimensional statistical methods (the Minimax-Linkage Prototype Clustering and LASSO) used in the GIBS algorithm can be found in Appendix A. The sketch of the GIBS algorithm is shown in the Table 1. The details of the GIBS algorithm can be found in the paper Zhu et al. (2020) [24]. An application can be found in Jarrow et al. (2021) [13].

For simplicity, denote $V_{1}(t)=B(t)$ as the money market account, and $V_{2}(t)$ as the market index. Most of the ETFs, $\Delta\bm{V}_{i}$, are correlated with $\Delta\bm{V}_{2}$, the market portfolio. Although this is not true for the other 4 Fama-French factors. Therefore, we first orthogonalize every other basis asset (excluding $\Delta\bm{V}_{1}$ and $\Delta\bm{V}_{2}$) to $\Delta\bm{V}_{2}$. By orthogonalizing with respect to the market return, we avoid choosing redundant basis assets and increase the accuracy of the fitting. Note that for Ordinary Least-Squares (OLS) regression, projection does not affect the estimation of $\Delta\bm{\hat{Y}}$, since it only affects the coefficients. However, in LASSO projection does affect the set of selected basis assets because it changes the magnitude of the coefficients before shrinking. Thus, we compute

$$\widetilde{\Delta\bm{V}}_{i}=(\bm{I}-P_{\Delta\bm{V}_{2}})\Delta\bm{V}_{i}=(\bm{I}-\Delta\bm{V}_{2}(\Delta\bm{V}^{\prime}_{2}\Delta\bm{V}_{2})^{-1}\Delta\bm{V}_{2}^{\prime})\Delta\bm{V}_{i}\qquad\text{where}\quad 3\leq i\leq p_{2}$$

(11)

where $P$ denotes the projection operator. Let

$$\widetilde{\Delta\bm{V}}=(\bm{V}_{1},\bm{V}_{2},\widetilde{\Delta\bm{V}_{3}},...,\widetilde{\Delta\bm{V}_{p_{2}}}).$$

(12)

Note that this is equivalent to the residuals after regressing other risk factors on $\Delta\bm{V}_{2}$.

The transformed ETF basis assets $\widetilde{\Delta\bm{V}}$ still contain highly correlated members. We first divide these basis assets into categories $C_{1},C_{2},...,C_{k}$ based on a financial characterization. Note that $C\equiv\cup_{i=1}^{k}C_{i}=\{1,2,...,p_{1}\}$. The list of categories with descriptions can be found in Appendix B. The categories are (1) bond/fixed income, (2) commodity, (3) currency, (4) diversified portfolio, (5) equity, (6) alternative ETFs, (7) inverse, (8) leveraged, (9) real estate, and (10) volatility.

Next, from each category we choose a set of representatives. These representatives should span the categories they are from, but also have low correlation with each other. This can be done by using the prototype-clustering method with a distance measure defined in Appendix A, which yield the “prototypes” (representatives) within each cluster (intuitively, the prototype is at the center of each cluster) with low-correlations.

Within each category, we use the prototype clustering methods previously discussed to find the set of representatives. The number of representatives in each category can be chosen according to a correlation threshold. This gives the sets $D_{1},D_{2},...,D_{k}$ with $D_{i}\subset C_{i}$ for $1\leq i\leq k$. Denote $D\equiv\cup_{i=1}^{k}D_{i}$. Although this reduction procedure guarantees low-correlation between the elements in each $D_{i}$, it does not guarantee low-correlation across the elements in the union $D$. So, an additional step is needed, in which is prototype clustering on $D$ is used to find a low-correlated representatives set $U$. Note that $U\subseteq D$. Denote $p_{2}\equiv\#U$.

Recall that $\widetilde{\Delta\bm{V}_{U}}$ means the columns of the matrix $\widetilde{\Delta\bm{V}}$ indexed by the set $U$. Since basis assets in $\widetilde{\Delta\bm{V}_{U}}$ are not highly correlated, a LASSO regression can be applied. Therefore, we have that

$$\bm{\widetilde{\beta}}_{i}=\underset{\bm{\beta}_{i}\in\mathbb{R}^{p},(\bm{\beta}_{i})_{j}=0(\forall j\in U^{c})}{\arg\min}\left\{\frac{1}{2n}\left\|\Delta\bm{Y}_{i}-\widetilde{\Delta\bm{V}_{U}}\bm{\beta}_{i}\right\|_{2}^{2}+\lambda\left\|\bm{\beta}_{i}\right\|_{1}\right\}$$

(13)

where $U^{c}$ denotes the complement of $U$. However, here we use a different $\lambda$ as compared to the traditional LASSO. Normally the $\lambda$ of LASSO is selected by cross-validation. However this will overfit the data as discussed in the paper Zhu et al. (2020) [24]. So here we use a modified version of the $\lambda$ selection rule and set

$$\lambda=\max\{\lambda_{1se},min\{\lambda:\#supp(\bm{\widetilde{\beta}}_{i})\leq 20\}\}$$

(14)

where $\lambda_{1se}$ is the $\lambda$ selected by the “1se rule”. The “1se rule” provides the most regularized model such that the error is within one standard error of the minimum error achieved by cross-validation (see [8, 19, 22]). Therefore, the set of basis assets selected is

$$S_{i}\equiv supp(\bm{\widetilde{\beta}}_{i}).$$

(15)

Next, we fit an Ordinary Least-Square (OLS) regression on the selected basis assets, to estimate $\bm{\hat{\beta}}_{i}$, the OLS estimator from

$$\Delta\bm{Y}_{i}=\Delta\bm{V}_{S_{i}}(\bm{\beta}_{i})_{S_{i}}+\Delta\bm{\epsilon}_{i}.$$

(16)

Since this is an OLS regression, we use the original basis assets $\Delta\bm{V}_{S_{i}}$ rather than the orthogonalized basis assets $\widetilde{\Delta\bm{V}}_{S_{i}}$. Note that $supp(\bm{\hat{\beta}}_{i})\subset S_{i}$. Since we are in the OLS regime, significance tests can be performed on $\bm{\beta}_{i}$. This yields the significant set of coefficients

$$S_{i}^{*}\equiv\{j:P_{H_{0}}(|\bm{\beta}_{i,j}|\geq|\bm{\hat{\beta}}_{i,j}|)<0.05\}\quad\text{where}\quad H_{0}:\text{True value}\,\,\bm{\beta}_{i,j}=0.$$

(17)

Note that the significant basis asset set is a subset of the selected basis asset set. In another words,

$$S_{i}^{*}\subseteq supp(\bm{\hat{\beta}}_{i})\subseteq S_{i}\subseteq\{1,2,...,p\}.$$

(18)

A sketch of the GIBS algorithm is shown in Table 1. Recall that for an index set $S\subseteq\{1,2,...,p\}$, $\Delta\bm{V}_{S_{i}}$ means the columns of the matrix $\Delta\bm{V}$ indexed by the set $S_{i}$.

We test the validity of the generalized APT by adding an intercept (Jensen’s alpha) and testing if the intercept is non-zero. A non-zero intercept implies that the securities are mispriced (i.e. the rejection of the existence of an equivalent martingale measure). Formally, we add an intercept term $\alpha_{i}$ to equation (16) and test the null hypothesis

$$H_{0}:\alpha_{i}=0\quad vs.\quad H_{a}:\alpha_{i}\neq 0.$$

(19)

Since using price differences removes the intercept, the intercept test has to done using prices

$$\bm{Y}_{i}=\alpha_{i}\mathbbm{1}_{n}+\bm{V}_{S_{i}}(\bm{\beta}_{i})_{S_{i}}+\bm{\epsilon}_{i}.$$

(20)

where $\mathbbm{1}$ is an $n\times 1$ vector with all elements 1, and $S_{i}$ are the basis assets selected by the GIBS algorithm in the AMF model, defined in Equation (15). For the FF5, the $S_{i}$ are the Fama-French 5-factors and the risk free rate.

Our initial idea was to fit an OLS regression on the selected basis assets for each company and then report the p-values for the significance of $\alpha_{i}$. However, we observed that the mma’s value $\bm{V}_{1}$ is highly correlated to the constant vector $\mathbbm{1}$ because the risk-free rate is close to (or equal to) 0 for a long time. Therefore, including both $\mathbbm{1}$ and $\bm{V}_{1}$ in the regression leads to the inverse of a nearly-singular matrix, which gives unreliable results. In this case, since the correlation is so large, even projecting out $\mathbbm{1}$ from $\bm{V}_{1}$ does not solve this problem. So we used a two-step procedure instead. First, we estimate the OLS coefficient $(\bm{\hat{\beta}}_{i})_{S_{i}}$ from the the non-intercept model

$$\bm{Y}_{i}=\bm{V}_{S_{i}}(\bm{\beta}_{i})_{S_{i}}+\bm{\epsilon}_{i}$$

(21)

and calculate the estimation of residuals

$$\bm{\hat{\epsilon}}_{i}=\bm{Y}_{i}-\bm{\hat{Y}}_{i}\quad\text{where}\quad\bm{\hat{Y}}_{i}=\bm{V}_{S_{i}}(\bm{\hat{\beta}}_{i})_{S_{i}}.$$

(22)

Then, we fit an intercept-only regression on the residuals

$$\bm{\hat{\epsilon}}_{i}=\alpha_{i}\mathbbm{1}+\bm{\delta}_{i}$$

(23)

and report the p-value for the significance of the intercept. Using this technique, we avoided the collinearity issue. The results show that for all time periods, for all stocks, we can not reject the null hypothesis in either AMF or FF5. In other words, there is no significant non-zero intercept for either AMF or FF5 for any time period and for any stock. This evidence provides a validation of the generalized APT and the use of the AMF and FF5 models.

For each time period, this section tests for the time-invariance of the multi-factors beta coefficients in a linear setting. For these tests we use the first order differences of the prices as described in equations (6) and (16). We use price differences to avoid autocorrelation and any non-stationarities in the price process. Here we only focus on the selected basis assets. In other words, we only test the time-invariance of $\beta_{i,j}$ where $j\in S_{i}$ and $S_{i}$ is the one defined in equation (15). Our null hypothesis for each stock $i$ is that “$H_{0}$: $\beta_{i,j}$ are time-invariant over the 4 years.”

Denote $\bm{h}=(h_{1},h_{2},...,h_{n})^{\prime}$ where $h_{i}=0$ for all rows related to first half of the time period and $h_{i}=1$ for all rows related to the second half of the time period. Testing for the significance of interaction of each basis asset with $\bm{h}$ is a way to test whether the coefficients are the same for the first and last two years. To be more specific, consider the regression model:

$$\Delta\bm{Y}_{i}=\Delta\bm{V}_{S_{i}}(\bm{\beta}_{i})_{S_{i}}+\left[\Delta\bm{V}_{S_{i}}\odot(\bm{h}\mathbbm{1}_{n}^{\prime})\right]\bm{\theta}_{i}+\Delta\bm{\epsilon}_{i}$$

(24)

Note that the sign “$\odot$” means the element-wise multiplication for two vectors. Here $\theta_{i,j}=0$ indicates that $\beta_{i,j}$ is time-invariant during the time period. Our null hypothesis becomes

$$H_{0}:\theta_{i,j}=0\;\;\text{for}\;\;\forall j\quad vs.\quad H_{1}:\exists j,\;\;\text{such that}\;\;\theta_{i,j}\neq 0.$$

(25)

An ANOVA test is employed to compare the model in equation (24) and (16). A p-value of less than 0.05 rejects the null hypothesis that the $\beta_{i,j}$’s are all time-invariant. We also want to control for the False Discovery Rate (FDR) (see [3]). The Benjamini-Hochberg (BH) [3] FDR adjusting procedure does not account for the correlation between tests, while the Benjamini-Hochberg-Yekutieli (BHY) [4] FDR adjusting method does. Since we may have correlation between the basis assets, we use the BHY method to adjust the p-values into the FDR Q-values and then report the percentage of stocks with Q-values less than 0.05 in Figure 1.

Figure 1 reports the percentage of stocks with time-varying beta using the time-invariance test in a linear setting for each time period. The y-axis is the start year of each time period and the x-axis is the end year of the time period. The percentage in each grid is the percentage of stocks with FDR Q-threshold 0.05 in the ANOVA test comparing the models in Equation (24) and (16). The larger the percentage is, the darker the grid will be. The upper heatmap is the result for the AMF model, while the bottom heatmap is the result for the FF5 model.

In the heatmaps in Figure 1 (and same below), all elements on the k-th skew-diagonal (see definition in Equation (9)) correspond to time periods of the same length, which is $k+3$ years. For example, the diagonal (0-th skew-diagonal) elements are related to the time periods with 3 years, the 1st skew-diagonal elements are of the time periods 4 years. Comparing the different skew-diagonals, we can see that the AMF model is very stable in all time periods less than 5 years. For most time periods no more than 5 years, only less than 5% companies has a least 1 time-varying $\beta$. In other words, for more than 95% of the companies, the $\beta$’s in the AMF model are time-invariant. However, for FF5 model, even some 3-year time periods are not stable, such as 2007 - 2009.

In the heatmaps in Figure 1 (and same below), all elements on the k-th skew-anti-diagonal (see definition in Equation (10)) correspond to time periods with the same “mid-year”. For example, the 1st skew-anti-diagonal elements are all related to time periods centered in the mid week of 2012, with different time lengths. By comparing different skew-anti-diagonals, we can compare if the stability pattern are the same for different mid-years. For AMF model, we can see that the percentage for a fixed time length does not change much for different mid-years. For example, all time periods of 4 years has time-varying percent less then 5%, which does not change much for different mid-years. However, the stability of the $\beta$’s for FF5 model depends highly on the mid-year, not just the length of the time period. FF5 is more volatile in the mid-years 2012 - 2013 and 2008 - 2009. FF5 can not capture the basis assets as accurately as the AMF, and the $\beta$’s change significantly during the financial crisis.

In general, the AMF is more stable than the FF5, which can be seen in Figure 2. The table in Figure 2 is the difference between the two tables in Figure 1 (AMF - FF5). For most time periods, the grid is blue, meaning that the AMF model is more stable than FF5 by having less a percentage of companies with time-varying $\beta$’s; sometimes the decrease is over 20%. FF5 is slightly more stable than AMF in only a few time periods, although both AMF and FF5 are quite stable, giving less than 5% of the companies with time-varying $\beta$’s. AMF performs much better than FF5 in all other time periods when the FF5 is unstable.

In summary, AMF outperforms the FF5 in terms of the stability in two ways. First, for all time periods, AMF is more stable (or at least equally stable) than FF5, i.e. AMF either gives more stable $\beta$’s than FF5, or gives equally stable $\beta$’s if FF5 is also stable. Second, the stability of the AMF model for each time length is more robust across all mid-years as compared to FF5. The stability of the AMF model only depends on the length of the time period, and not the starting or ending year, while the stability of the FF5 model depends on the mid-year. This implies that the AMF performed well during the financial crisis.

In this section we test if including more basis assets improves the fit in each time period. Since the number of ETFs increased over time, by focusing on the second half of the time period, we have more ETFs available compared to the beginning. We want to test if the newly introduced ETFs in the second half of the time period provides a better fit for the AMF model.

Formally, for a time period $[t_{a},t_{b}]$, let $t_{mid}=(t_{a}+t_{b})/2$. We divide $\Delta\bm{Y}_{i}$ and $\Delta\bm{V}$ into two parts, one for the time period $[t_{a},t_{mid})$ and the second for the time period $[t_{mid},t_{b})$, where

$$\Delta\bm{Y}_{i}=\begin{pmatrix}\Delta\bm{Y}_{i,a}\\ \Delta\bm{Y}_{i,b}\end{pmatrix},\quad\Delta\bm{V}=\begin{pmatrix}\Delta\bm{V}_{a}\\ \Delta\bm{V}_{b}\end{pmatrix}.$$

(26)

We first derive the basis assets set $S_{i}$ in Equation (15) using the GIBS algorithm on the whole time period. Then, we fit an OLS regression on the second half

$$\Delta\bm{Y}_{i,b}=(\Delta\bm{V}_{b})_{S_{i}}(\bm{\gamma}_{i})_{S_{i}}+\Delta\bm{\epsilon}_{i,b}$$

(27)

and obtain the estimated coefficients $(\bm{\hat{\gamma}}_{i})_{S_{i}}$. The residuals are

$$\Delta\bm{\hat{\epsilon}}_{i,b}=\Delta\bm{Y}_{i,b}-(\Delta\bm{V}_{b})_{S_{i}}(\bm{\hat{\gamma}}_{i})_{S_{i}}.$$

(28)

We fit an AMF model with the GIBS algorithm again, using the residuals $\Delta\bm{\hat{\epsilon}}_{i,b}$ as our new dependent variables, using all the basis assets available for the time period $[t_{mid},t_{b})$, except for the basis assets already selected in $S_{i}$. This AMF model on the residuals will provide another selected set of basis assets $S_{i,b}$. If $S_{i,b}\neq\emptyset$, we merge the two sets together

$$S_{i,union}=S_{i}\cup S_{i,b}.$$

(29)

Note that since we remove all the basis assets in $S_{i}$ in our GIBS fitting of the residuals, $S_{i}\cap S_{i,b}=\emptyset$. Continuing, we fit another OLS regression on the second half of the data using the basis assets $S_{i,union}$

$$\Delta\bm{Y}_{i,b}=(\Delta\bm{V}_{b})_{S_{i,union}}(\bm{\gamma}_{i})_{S_{i},union}+\Delta\bm{\epsilon}_{i,b}.$$

(30)

Finally, we use an ANOVA test to compare the models in Equation (27) and Equation (30). For each time period, this test is performed on all stocks yielding a list of p-values. As before, we adjust for FDR using the BHY [4] method and count the number of companies with FDR Q-values less than 0.05. The percentages are reported in Figure 3.

Note that if for $i$-th stock the set $S_{i,b}=\emptyset$, there will not be p-value for this stock since the two models in Equation (27) and (30) are the same. So this stock will not be counted as a company with a FDR Q-value less than 0.05. However, when presenting the percentage, we use the total count of companies available in that time period in the denominator. This generates more conservative percentages.

For the FF5 model, $S_{i}$ is the FF5 5-factors and the risk free rate. All the remaining steps in the procedure are the same. The results for FF5 residuals provide a comparison. From Figure 3 we see that for all time periods, the second half gives a significantly better fit with the new basis assets for around 15% of the stocks in the AMF model. Comparing this result with Figure 1, although there are stocks that have constant $\beta$’s, their risks can be better fitted using the new basis assets. This provides an interesting insight. More basis assets provides better fitting of the $\epsilon$ errors, although the affects of the old basis assets are still the same.

In summary, comparing the results for the AMF and FF5 residuals, it is clear that AMF uses more basis assets and leaves less information in the residuals than foes FF5. From Figure 4 we see that the percentage of stocks that are better fitted in the second half of the time period is always less for AMF than it is for FF5.

This section reports the In-Sample and Out-of-Sample goodness-of-fit tests. Since the AMF selects more basis assets than the FF5, it is important to study overfitting. The results show that the AMF model is more powerful and less vulnerable to overfitting than is the FF5, because the AMF achieves a better in-sample Adjusted $R^{2}$ (see [20]) and a better Out-of-Sample $R^{2}$ (see [6]).

The Adjusted $R^{2}$ (see [20]) is a measure of goodness of fit similar to the normal $R^{2}$ but adjusted for the number of parameters to penalize on overfitting. The results are shown in Figure 5 and the difference of the Adjusted $R^{2}$ between AMF and FF5 is shown in Figure 6. It is clear that for all time periods, AMF increases the Adjsuted $R^{2}$ by around 0.1 compared to FF5, which is around a $0.1/0.25=40\%$ increment.

The Out-of-Sample $R^{2}$ (see [6]) is a measure of a model’s prediction power. It can be used to check whether a model is overfitting. For each time period, we calculate the Out-of-Sample $R^{2}$ for AMF and FF5 for the half-year after the end of that period. The results are in Figure 7 and the differences of the Out-of-Sample $R^{2}$ between the two models are in Figure 8. We see that the prediction power of FF5 fades dramatically with time, while the prediction power of AMF is stable, and much better than the FF5. The Out-of-Sample $R^{2}$ for AMF is around 0.47, while that for FF5 is around 0.3 in early years and 0.2 in recent years. In recent years, the prediction power of the AMF model is almost twice as large as that of FF5. Therefore, it is clear that the basis assets selected by the AMF model are not overfitting, but achieving better in-sample and out-of-sample goodness-of-fit.

We can also test time-invariance based on thr non-linear Generalized Additive Model (GAM). This is a more strict test, since this tests not only for the time-invariance of the $\beta$’s, but also their linearity. We first fit a Time-Varying Coefficient model as a special case of the GAM. The equation can be written as:

$$\Delta\bm{Y}_{i}=\Delta\bm{V}_{S_{i}}[\bm{\beta}_{i}(t)]_{S_{i}}+\Delta\bm{\epsilon}_{i}$$

(31)

where $t$ is the time. Note that the only difference between Equation (31) and Equation (16) is the second allows the $\beta$’s to be functions of time $t$. The GAM model estimate each $\beta$ as a combination of splines or kernels with regard to $t$. This can be done by the gam() function in the R package mgcv.

Next, we perform a ANOVA test between the GAM model in Equation (31) and the linear model in Equation (16) where each $\bm{\beta}_{i}$ are constants over time. This yields a p-value for each stock. As before, we adjust the p-values using the BHY [4] method to account for FDR. We report the percentage of stocks with Q-values less than 0.05 in Figure 9.

Figure 9 reports the percentage of stocks with time-varying beta using the time-invariance test in a non-linear GAM setting for each time period. The y-axis is the start year and the x-axis is the end year of each time period. The percentage in each grid is the percentage of stocks with FDR Q-threshold 0.05 in the ANOVA test that compares the models in Equation (31) and (16). The larger the percentage, the darker the grid will be. The upper heatmap is for the AMF model, while the bottom heatmap is for the FF5 model.

By comparing the different skew-diagonals, we see that both models are more stable in shorter time periods. The AMF outperforms the FF5 in all the time periods, as seen in Figure 10. Figure 10 gives the difference between the percentages across the two heatmaps in Figure 9 (AMF - FF5). All the grids are blue, indicating that AMF is more stable than FF5 in all time periods.

However, comparing the results from GAM and the results from the previous Section 4.2, for any time period and both AMF and FF5 models, more companies are shown to have time-varying $\beta$’s in the GAM test, although the AMF still outperforms the FF5. This does not necessarily mean that the $\beta$’s are time varying in all time periods, since it is easy for GAM to overfit, especially in shorter time periods. Indeed, the number of observations for 3 year time periods is $n=156$. For the FF5 model, there are 5 basis assets. For the AMF model, there are more basis assets selected. For each basis asset, the GAM model selects some splines or kernels, say 10 splines, which can easily expand the dimension of parameters to $p=5\times 10=50$, which is already too large for a regression with $n=156$ observations. The number of parameters becomes too close to the number of observations. In addition, these new variables may be highly correlated, making the fitt more unstable. This can result in the GAM overfitting. Therefore, the testing based on the GAM model in this section is explorative. Considering this high-dimensional issue, penalization and constraints need to be introduced to traditional GAM for our application. This extension is left for future research.