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Slide 1 - Regularization Models Why you should avoid them Gaetan Lion, December 9, 2021 1
Slide 2 - 2 What is Regularization? … OLS Regression + Penalization LASSO: MIN[Sum of Squared Residuals + Lambda(Sum of Absolute Regression Coefficients)] Ridge Regression: MIN[Sum of Squared Residuals + Lambda(Sum of Squared Regression Coefficients)]
Slide 3 - 3 Showing the OLS term (yellow) vs. Penalization term (orange) LASSO: MIN[Sum of Squared Residuals + Lambda(Sum of Absolute Regression Coefficients)] Ridge Regression: MIN[Sum of Squared Residuals + Lambda(Sum of Squared Regression Coefficients)] Lambda is simply a parameter, a value, or a coefficient if you will. If Lambda = 0, the LASSO or Ridge Regression = OLS Regression If Lambda is pretty high, the penalization is more severe. And, the variables regression coefficients will either be zeroed out (LASSO) or very low (Ridge Regression).
Slide 4 - 4 Given that regularization should be conducted with standardized coefficient, such a model structure that penalizes high variable coefficients also penalizes variable statistical significance and variable influence on the behavior of the dependent variable. That’s not a robust modeling concept.
Slide 5 - 5 Capturing a model forecasting accuracy: A LASSO Regularization model that worked (left graph) vs. one that did not (right graph). These graphs represent LASSO models forecasting accuracy or error given different penalization Lambda levels. The X-axis represents Lambda levels. As Lambda rises going to the right, the penalty factor is stronger. And, the variables regression coefficients are lowered and even zeroed out. The higher X-axis values represent the number of variables left in the LASSO model. So, the number of variables decreases as you go further to the right with rising penalty (that’s how LASSO models work). The Y-axis discloses the cross validation Mean Squared Error as a test of a model forecasting accuracy.
Slide 6 - 6 Model overfitting Model under-fitting This LASSO model is successful. It started with 46 variables (way too many variables). The LASSO model far improved forecasting accuracy (lower MSEs) by eventually keeping only one single variable in the model (out of the 46 original one). This LASSO model is very successful. It starts with just 5 variables. And, the minute it either shrinks those coefficients or eliminates variables (through higher Lambda penalization), the model MSEs quickly rises. This is a case of model under-fitting
Slide 7 - 7 Maintaining explanatory logic of a model… or not: Ridge Regression The coefficient path graphs given different level of Lambdas disclose if the explanatory logic of a model is maintained or not. Notice that the Lambda penalization on the left graph increases from left to right. On the right hand graph, penalization increases from right to left (both graph directions are common depending on what software you use). This Ridge Regression is very successful in maintaining the explanatory logic of the model. At any Lambda level, the variables coefficients maintain their relative weight and directional sign (+ or - ). This Ridge Regression fails in maintaining the explanatory logic of the model. At any level of Lambda, the coefficients relative weight drastically change. They even often flip sign (+ or -).
Slide 8 - 8 Maintaining explanatory logic of a model… or not: LASSO Good Bad The comments on the previous slide are applicable here. Just note, the visual difference. A Ridge Regression does not readily completely zero out the coefficients. Meanwhile, a LASSO model does that resulting in truncated paths towards the Zero line, as variables get eliminated with rising Lambda penalty.
Slide 9 - 9 What a good Regularization Model should look like Improved forecasting accuracy Maintained explanatory logic Unless a regularization model fares well on both components (forecasting accuracy, explanatory logic), a Regularization model can’t be deemed successful.
Slide 10 - 10 When Regularization may work vs. not OLS with proper fit Regularization causes under-fitting Overfit model Regularization reduces overfitting A model with a lot of splices, nodes, and related polynomials can often be overfit. In such a case, Regularization can reduce model overfitting. An OLS regression is often not overfit to begin with. And, in such circumstances, a Regularization will flatten the slope of the regression trend line, and causes model under-fitting.
Slide 11 - 11 Doing a specific Ridge Regression example
Slide 12 - 12 Starting with an OLS Regression to estimate Real GDP growth We constructed an OLS Regression to fit Real GDP quarterly growth since 1959 using a pool of 17 prospect independent variables with up to 4 quarter lags for a total of 85 different prospect variables (17 x 4 = 68 + 17 = 85). We came up with a pretty good explanatory model with 7 variables, including: Labor force Lag 1 quarter (laborL1) Velocity of money (M2/GDP) M2 Lag 1 quarter S&P 500 level Lag 1 quarter Fed Funds rate Lag 3 quarter and Lag 2 quarter 10 Year Treasury bill Lag 1 quarter. Each variable was fully detrended (on either a quarterly % change basis or a First Difference basis, as is most relevant). And, each of those detrended variables were standardized (average = 0, standard deviation = 1).
Slide 13 - 13 Regularizing this OLS model -> Model under-fitting Output using R glmnet package The above is a picture of a failed regularization model. The best model is pretty much the OLS model when Lambda is close to Zero. The minute Lambda increases a bit, the MSE rapidly increases showing a deterioration in forecasting accuracy. The Fraction Deviance Explained is very much the same as R Square. The minute the Ridge Regression shrinks a bit the coefficients, the R Square equivalent drops fairly rapidly.
Slide 14 - 14 Very different Ridge Regression coefficient shrinkage given specific Lambda penalization with R glmnet vs. other software packages Whether you use the R MASS, R penalized, or Python sk learn packages, you get nearly the exact same coefficient shrinkage given a Lambda level (left graph). And, that shrinkage is close to Zero indicating that the original OLS regression was not overfit. With the R glmnet package you get drastically more coefficient shrinkage. But, as indicated on the previous slide, this large shrinkage also corresponds to very pronounced model under-fitting.
Slide 15 - 15 Another look at the dramatic R glmnet coefficient shrinkage… this time in % For all the mention package, regardless of the Lambda level (up to 5), the shrinkage was pretty small (always much less than – 7.0%). With the R glmnet package, the coefficient shrinkage is pretty dramatic and often reaches – 80% or more. A coefficient that shrinks by more than – 100% switches signs. This is the case with the 10 Year Treasury rate (t10L1).
Slide 16 - 16 Doing variable selections with stepwise-forward and LASSO We will use the same data set of 85 prospect independent variables to fit Real GDP growth.
Slide 17 - 17 Stepwise-forward using R olsrr package
Slide 18 - 18 Variable selection using LASSO models When conducting Ridge Regression, glmnet was the outlier with very different results using the same Lambda penalty level. Now with LASSO, somehow glmnet generates the same results as Python sk learn given the same Lambda level. And, it is the R penalized package that is the outlier. We used Lambda level so as to approach a number of selected variables that be similar to the stepwise methodology (12 selected variables).
Slide 19 - 19 Comparing the models based on variables’ influence or materiality The LASSO models select a few more variables. But, far fewer of them are “material.” By, material we mean an independent variable that has an absolute standardized coefficient > 0.1. For the stepwise-forward model, 50% of the selected variables have an absolute standardized coefficient > 0.1. For the LASSO models, with sk learn and glmnet, only 2 of them have a “material” coefficient. With the R penalized package 5 out of 17 of them, or 29.4% have a material coefficient. The sk learn and glmnet LASSO models are left with very little explanatory logic as their fit relies primarily on just two variables (out of 14, the other 12 are pretty much immaterial with incredibly low coefficients).
Slide 20 - 20 How about Multicollinearity The stepwise selection model does have some multicollinearity. Either the Velocity or M2/GDP variable should be removed from the model. The sk learn and glmnet LASSO models have resolved multicollinearity by selecting only two variables with “material” coefficients. And, these two variables (Velocity and Labor Lag 1) are not excessively correlated. The R penalized have a similar multicollinearity profile as the stepwise selection model. Because of the LASSO coefficient shrinkage, the related coefficients are a bit lower. And, it may abate multicollinearity somewhat… but most probably not entirely. Coefficients can be relatively smaller, but nearly as unstable because of multicollinearity.
Slide 21 - 21 The glmnet LASSO model is not successful in improving forecasting The MSE line remains pretty flat when it includes the majority of the variables within this variable selection process. It improves marginally, when it applies still very low Lambda levels and shrinks variable selection down to 30 variables. However, further to the right, the MSEs rise rapidly when the model includes less than 22 variables. Notice that all Lambda considered are for the most part very small as they are all under 1. What is true for the R glmnet LASSO model is also true for the Python sk learn model since they pretty much replicate each other results on this one count.
Slide 22 - 22 The penalized LASSO model is inconsistent As you increase Lambda from 3 to 10, coefficients get increasingly shrunk and many get zeroed out. The resulting number of variables selected declines from 26 when Lambda is 3 to 17 when Lambda is 10. But, notice how some variables are newly selected when Lambda increases. For instance: ffL2 gets selected for the first time when Lambda increases to 10; M2/GDP Lag 1 gets selected for the first time when Lambda increases to 4; 5 year Treasury Lag 3 gets selected for the first time when Lambda increases to 4; and Velocity Lag 3 gets selected for the first time when Lambda increases to 10. None of the above seem right for a LASSO regression. Variables should not get newly selected when Lambda rises.
Slide 23 - 23 How to better resolve model specification issues not well addressed by Regularization
Slide 24 - 24 How to diagnose model overfitness Check the model Adjusted R Square that penalizes for adding variables; Check the model Information Criteria (AIC, BIC) that also penalize for adding variables; Conduct cross validation. An overfit model will have a better historical fit (lower error) than another model, but will generate larger cross validation errors.
Slide 25 - 25 How to reduce or eliminate model overfitness Just eliminate the variables that have the least impact on the model fit and are associated with the least improvement in RMSE reduction. For instance, the stepwise-forward procedure we ran earlier selected 12 variables based on p-value thresholds (=< 0.10). But, the first 6 variables contribute the majority of the information. The other 6 are likely to contribute to model overfitness.
Slide 26 - 26 Multicollinearity: statistical significance This is the problem that does not exist. Let me explain. When two independent variables are highly correlated, it is supposed to impair their respective statistical significance. And, when such variables are highly correlated and characterized by a Variance Inflation Factor (VIF) of 5 or 10, such variables are deemed multicollinear and one of them should be removed. But, VIF is an “after-the-fact” test. Within the model we already have assessed that the variables are statistically significant. If we removed the one multicollinear variable it would only improve the statistical significance of the other related remaining multicollinear variable beyond a mandated threshold of statistical significance. In summary, this improvement is superfluous. Do you care if a t-stat of a variable is 3 or 6? Let’s take an example. A multicollinear variable has a t-stat of 2, a p-value of 0.05, and a VIF of 5. If we remove its partnering-multicollinear variable, its t-stat could potentially double to 4. But, this is a superfluous improvement since a t-stat of 2 is already statistically significant. The Standard Error of the regression coefficient multiple is equal to the square root of the Variance Inflation Factor (VIF).
Slide 27 - 27 Multicollinearity: coefficient instability Ok, that is a far more pressing problem. To test for that run a set of Rolling Regressions where you cut out a rolling window of data (let’s say 5 years or 20 quarters of data) and observe how the variables coefficients move over time. By doing so, you will readily identify the variable coefficients that are unstable. Coefficient instability can be caused by many different things besides multicollinearity. It often is caused by instability (outliers) within the independent variables. In such circumstances some instability within the variable coefficients is deemed acceptable. However, if two variables are multicollinear and their respective coefficients are unstable, removing one of those variables should help the coefficient stability of the variable that remains in the model.
Slide 28 - 28 Coefficient instability… another solution: Robust Regression to outliers There is a very interesting family of linear regressions that are robust to outliers. They are helpful in reducing coefficient instability that is associated with volatility, change of regime, and other divergent movements within the independent variables and even within the dependent variable. In other words, these regressions are robust to outliers of all kinds. The most common ones will be described shortly. But, first let’s look at the different types of outliers as diagnosed with an Influence Plot.
Slide 29 - 29 Understanding & Uncovering Outliers Cook's D (bubble size) It measures the change to the estimates that results from deleting an observation. It combines Outlierness on both the y- and x-axes. Threshold:> 4/n Studentized Residuals (y-axis) Dependent variable outliers Large error. Unusual dependent variable value given independent variable’s input. Threshold: + or - 2. This means an actual data point is two standard errors (scaled to a t distribution) away from the regressed line. Hat-Leverage (x-axis) Independent variable outliers Leverage measures how far an independent variable deviates from its Mean. Threshold: >(2k + 2)/n
Slide 30 - 30 Influence Plot: understanding Outliers influence or impact The outliers in the top right-hand and bottom right-hand sections (green zones) are the most influential. They have residuals that are more than 2 standard errors (adjusted for t distribution) away from the actual value. And, they also have high Hat-values (x variable outlier). Their resulting overall influence as measured by Cook’s D value (bubble size) are the largest.
Slide 31 - 31 Robust Regression Methods M-estimation. The M stands for “maximum likelihood type.” Also, called Iteratively Reweighted Least-Squares (IRLS). The method is resistant to Y outliers (Studentized residuals) but not X outliers (Leveraged points). This method is efficient and has a reasonably good regression fit. There is two M-estimation version. The first one is called Huber M-estimation. The second one is M-estimation bisquare. The bisquare version may have more continuous weighting of observations. Difference between the two is often immaterial. S-estimation. This method finds a line (plane or hyperplane) that minimizes a robust estimate of the scale (why it is called S-est.) of the residuals. This method is resistant to both Y and X outliers. But, it is less efficient. MM-estimation. This method combines the efficiency of M-estimation with the resistance to both Y and X outliers. It has also two versions (traditional and bisquare). Difference often not material. L1 Quantile Regression. This method is resistant to both Y and X outliers by regressing estimates to the Median instead of the Mean (like in OLS). Thus, regression coefficients are less affected by outliers. It can withstand up to 29% reasonably bad data points (John Fox, 2010). Computation relies on linear programming, and don’t always converge on a perfect solution (Median of estimates often different from Median of actuals). Nevertheless, it is reasonably efficient. Least trimmed squares (LTS). This method is resistant to both Y and X variable outliers. It minimizes the sum of the square of the residuals, just like OLS, but only on little more than half of the observations* away from the tails. However, it can be much less efficient. Also, there is no formula for coefficient standard errors. So, variables stat. sign. is tough to evaluate. *It is slightly more than half and is estimated at: m = n/2 + (k + 2)/2. (Source: Robust Regression in R, John Fox & Sanford Weisberg, 2010).
Slide 32 - 32 Robust Regression Methods Summary MM-estimation and L1 Quantile Regression are among the preferred Robust Regression methods to deal with outliers given their versatility and strengths on all dimensions.
Slide 33 - 33 Considerations As reviewed, Regularization can often introduce numerous model weaknesses as outlined on the fourth slide including: Model under-fitting; Poor forecasting accuracy; and weakened explanatory logic. Additionally, Regularization can be highly unstable or inconsistent across software platforms resulting in divergent penalization levels depending on what software you use. All the model issues that Regularization attempt to address can be resolved in more reliable ways. Often, eliminating superfluous variables that can be readily identified (see slide 25) will resolve most issues. You can also use Robust Regression to improve coefficient stability.
 

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