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Home Price Models Presentation Transcript

Slide 1 - Housing Price Models Gaetan Lion, May 23, 2022
Slide 2 - Introduction 2 Objectives My first objective was to model housing prices at the county level using explanatory demographic variables. My second objective was to benchmark four different models varying in complexity from simple linear regressions to more complex Deep Neural Network (DNN) models. Data I used county level home price data from Zillow (“zestimates”) and I used, tested, and selected demographic variables from the GEOFRED data in order to estimate the mentioned county-level home price zestimates. Some of the demographic variables at GEOFRED had data on up to close to 3,150 counties. The Zillow county level data had data on about 2,850 counties. When eliminating missing data on any of the tested variables, I ended up with a data set of over 2,500 counties. Variable transformation All variables are standardized so as to be on the same scale. Software R neuralnet package
Slide 3 - The selected variables 3 We considered many other demographic variables at GEOFRED. But, many were missing too many county-data points. Others were associated with correlations or regression coefficients that were either not statistically significant or of the wrong sign. The first 7 independent variables were selected as the best ones to construct an explanatory model. The 8th one (population change) was selected to construct a parsimonious predictive model.
Slide 4 - 4 The two Linear Regression Models OLS Long OLS Short This is an explanatory model that captures many socioeconomic dimensions : income, education, innovation, behavior, single motherhood, homeownership, and commute time. This is a parsimonious model that generates the same Goodness-of-fit with only 3 variables instead of 7. Remember all the variables are standardized. So, the regression coefficients are indicative of the relative weight of each variable. The derived coefficients were associated with using the entire data set.
Slide 5 - 5 The two Deep Neural Network Models DNN Soft Plus. 2 hidden layers (3, 2) DNN Logit. 2 hidden layers (4, 2) The DNN Soft Plus uses a more advanced smooth Rectified Linear Unit activation function called Soft Plus (See Appendix section). It is associated with two hidden layers. The first one with 3 neurons, and the second one with 2 neurons. This DNN Logit uses an older activation function: Sigmoid. The latter is a Logit Regression. This model structure had no problem converging towards a solution. However, the Sigmoid activation function is associated with coefficient compression issue when using more than one hidden layer (See Appendix).
Slide 6 - 6 DNN Soft Plus Convergence Issue DNN Soft Plus. 2 hidden layers (3, 2) DNN Logit. 2 hidden layers (4, 2) For the DNN Soft Plus model to converge towards a solution, we had to prune down the first layer from 4 neurons down to 3. And, we also had to increase the error threshold for the partial derivatives from 0.1 for the DNN Logit to 0.3 for the DNN Soft Plus model. As a result, when using the whole data, the DNN Soft Plus error at 447.5 is more than twice as large as for DNN Logit (189). And, the DNN Soft Plus needed 63% more steps (41,652 vs 25,521) to converge towards a solution.
Slide 7 - 7 Fitting the entire data set. The DNN Logit model is the clear winner The scatter plots top right hand quadrant defined by the red and green dashed lines show the homes with zestimates > $1 million. The DNN Logit models fit the zestimates > $1 million perfectly. The other three models do not fit well the > $1 million data points.
Slide 8 - 8 Fitting the entire data set. The DNN Logit model is the clear winner. Part II On all Goodness-of-fit measure, the DNN Logit model is way superior to the other three. It was expected since the DNN Logit could exploit non-linear relationships that the OLS models could not. Also, the DNN Logit model converged towards a solution with a much lower error than the DNN Soft Plus. Technical notes: When calculating the standard error, we assumed for simplicity, that each model had the same degree of freedom of 1. Given the large sample (> 2,500), this assumption did not affect the result much. The standard error was transformed from standardized units to nominal home values in $000. The error reduction is calculated by comparing the standard error of the model with the standard deviation of the dependent variable (which would be the standard error of a naïve model using the average of Y as a single estimate. Let’s say a model has a standard error of 5, and Y has a standard deviation of 10. The error reduction = 5/10 -1 = - 50%.
Slide 9 - 9 When we test the models, the DNN Logit performance is mediocre After using the total data, we tested the models twice using the following sample segmentations: Train 80% (learning sample) and Test (new data) 20%; Train 50%, Test 50%. When you look at all the Goodness-of-fit measures for the predictions in Test 20% and Test 50%, the DNN Logit performance falls abruptly. And, it is not any better, and at times worse, than the other three models.
Slide 10 - 10 Test 20% (new data) predictions scatter plots
Slide 11 - 11 Test 50% (new data) predictions scatter plots
Slide 12 - 12 A closer look at the DNN Logit (80%/20%) performance In training (80%), the model fit the data very well, including near perfect fit of the > $1 million homes. In the test (20%) predictions, there were 3 homes near $1 million, and the model was way off on all 3.
Slide 13 - 13 A closer look at the DNN Logit (50%/50%) performance Same situation as for the 80/20 testing. The perfect fit in training on the homes > $1 million did not help in predicting in testing similar homes > $1 million.
Slide 14 - 14 A perfect representation of overfitting … the DNN Logit model During training, the DNN Logit model gives you the illusion that it has captured very precise non linear relationships to perfectly fit the homes > $1 million (left graph). But, in the testing (right graph) this same model is unable to predict similar homes > $1 million. Thus, during the training the DNN Logit model really fit random noise much more than any true non linear relationships.
Slide 15 - 15 Overfitness within OLS vs DNN models The DNN Logit model has a much superior fit in training or when fitting using the whole data. But, is less accurate in prediction. Again, that is a classic definition of model overfitting. It overfits on random outliers using non linear DNN fitting capabilities that do not reflect true non linear relationships. The OLS models have reasonably equal performance in fitting actual data vs. in predicting new data (test). Given that, they are way less overfit than the DNN models (especially the DNN Logit one).
Slide 16 - 16 For predicting home prices, OLS Short is much better than DNN Logit OLS Short DNN Logit With just 3 variables, the OLS Short model predicts better than the DNN Logit with 7 variables and two hidden layers (4, 2). Also, OLS regression math is fast and closed form. DNN math is just the opposite.
Slide 17 - 17 For explaining home prices, OLS Long is much better than DNN Logit OLS Long DNN Logit For explanatory purpose, the OLS Long model is more transparent than the DNN Logit. OLS Long allows you to directly compare the relative weight of each sociodemographic factors. Meanwhile, the DNN Logit is opaque. And, its complexity is associated with more random noise than true explanatory power.
Slide 18 - 18 We did not speak much about the DNN Soft Plus model … … that’s because it was neither here nor there. It pretty much replicated the performance of the OLS models. And, it did that in the most burdensome and opaque way possible (these characteristics are rather typical of DNNs). In view of the above, right off the bat you would not choose it over the OLS models. By contrast, the DNN Logit model seemed most promising in training, as it was far superior to the other models. But, when conducting testing, it turned out that the DNN Logit was just way overfit.
Slide 19 - 19 A quick word about DNNs Activation Functions Appendix Section
Slide 20 - 20 Common DNNs Activation Functions Until around 2017, the preferred DNN activation function was the Sigmoid or Logistic one as it had an implicit probabilistic weight to a Yes or No loading of a node or neuron. However, soon after the Rectified Linear Unit (ReLU) became the preferred DNN activation function. We will advance that SoftPlus, also called smooth ReLU, should be considered a superior alternative to ReLU. See further explanation on the next slide.
Slide 21 - 21 The Sigmoid or Logistic Activation Function There is nothing wrong with the Sigmoid function per se. The problem occurs when you take the first derivative of this function. And, it compresses the range of the values by 50% (from 0 to 1, to 0 to 0.5 for the first iteration). In iterative DNN models, the output of one hidden layer becomes the input for the sequential layer. And, this 50% compression from one layer to the next can generate values that converge close to zero. This problem is called the “vanishing gradient descent.” We will see that in our situation, this problem is not material.
Slide 22 - 22 ReLU and smooth ReLU or SoftPlus Activation Functions SoftPlus appears superior to ReLu because it captures the weights of many more neurons’ features, as it does not zero out any such features with an input value < 0. Also, it generates a continuous set of derivatives values ranging from 0 to 1. Instead, ReLu derivatives values are limited to a binomial outcome (0, 1).
 

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