), government policies (prediction of growth rates for income, inflation, tax revenue, etc.) On the other hand, in smaller samples \(\widehat{Y}\) performs better than \(\widehat{Y}_{c}\). Because \(\exp(0) = 1 \leq \exp(\widehat{\sigma}^2/2)\), the corrected predictor will always be larger than the natural predictor: \(\widehat{Y}_c \geq \widehat{Y}\). \]
We have examined model specification, parameter estimation and interpretation techniques. &= 0
and let assumptions (UR.1)-(UR.4) hold. \], \[
&=\mathbb{E} \left[ \mathbb{E}\left((Y - \mathbb{E} [Y|\mathbf{X}])^2 | \mathbf{X}\right)\right] + \mathbb{E} \left[ 2(\mathbb{E} [Y|\mathbf{X}] - g(\mathbf{X}))\mathbb{E}\left[Y - \mathbb{E} [Y|\mathbf{X}] |\mathbf{X}\right] + \mathbb{E} \left[ (\mathbb{E} [Y|\mathbf{X}] - g(\mathbf{X}))^2 | \mathbf{X}\right] \right] \\
The key point is that the confidence interval tells you about the likely location of the true population parameter. We will examine the following exponential model:
\mathbb{C}{\rm ov} (\widetilde{\mathbf{Y}}, \widehat{\mathbf{Y}}) &= \mathbb{C}{\rm ov} (\widetilde{\mathbf{X}} \boldsymbol{\beta} + \widetilde{\boldsymbol{\varepsilon}}, \widetilde{\mathbf{X}} \widehat{\boldsymbol{\beta}})\\
\left[ \exp\left(\widehat{\log(Y)} - t_c \cdot \text{se}(\widetilde{e}_i) \right);\quad \exp\left(\widehat{\log(Y)} + t_c \cdot \text{se}(\widetilde{e}_i) \right)\right]
\[
The Statsmodels package provides different classes for linear regression, including OLS. Let’s use statsmodels’ plot_regress_exog function to help us understand our model. \]
Parameters: exog (array-like, optional) – The values for which you want to predict. \mathbb{V}{\rm ar}\left( \widetilde{\boldsymbol{e}} \right) &=
(415) 828-4153 toniskittyrescue@hotmail.com. We will show that, in general, the conditional expectation is the best predictor of \(\mathbf{Y}\). Linear regression is a standard tool for analyzing the relationship between two or more variables. \mathbf{Y} | \mathbf{X} \sim \mathcal{N} \left(\mathbf{X} \boldsymbol{\beta},\ \sigma^2 \mathbf{I} \right)
Prediction intervals must account for both: (i) the uncertainty of the population mean; (ii) the randomness (i.e. scatter) of the data. Next, we will estimate the coefficients and their standard errors: For simplicity, assume that we will predict \(Y\) for the existing values of \(X\): Just like for the confidence intervals, we can get the prediction intervals from the built-in functions: Confidence intervals tell you about how well you have determined the mean. statsmodels.regression.linear_model.OLSResults.conf_int ... Returns the confidence interval of the fitted parameters. We again highlight that \(\widetilde{\boldsymbol{\varepsilon}}\) are shocks in \(\widetilde{\mathbf{Y}}\), which is some other realization from the DGP that is different from \(\mathbf{Y}\) (which has shocks \(\boldsymbol{\varepsilon}\), and was used when estimating parameters via OLS). \begin{aligned}
predstd import wls_prediction_std # carry out yr fit # ols cinv: st, data, ss2 = summary_table (ols_fit, alpha = 0.05) \mathbb{E} \left[ (Y - \mathbb{E} [Y|\mathbf{X}])^2 \right] = \mathbb{E}\left[ \mathbb{V}{\rm ar} (Y | X) \right]. \end{aligned}
Y = \beta_0 + \beta_1 X + \epsilon
applies to WLS and OLS, not to general GLS, that is independently but not identically distributed observations \end{aligned}
We estimate the model via OLS and calculate the predicted values \(\widehat{\log(Y)}\): We can plot \(\widehat{\log(Y)}\) along with their prediction intervals: Finally, we take the exponent of \(\widehat{\log(Y)}\) and the prediction interval to get the predicted value and \(95\%\) prediction interval for \(\widehat{Y}\): Alternatively, notice that for the log-linear (and similarly for the log-log) model:
\], \[
\], \[
The confidence interval is a range within which our coefficient is likely to fall. \], \(\epsilon \sim \mathcal{N}(\mu, \sigma^2)\), \(\mathbb{E}(\exp(\epsilon)) = \exp(\mu + \sigma^2/2)\), \(\mathbb{V}{\rm ar}(\epsilon) = \left[ \exp(\sigma^2) - 1 \right] \exp(2 \mu + \sigma^2)\), \(\exp(0) = 1 \leq \exp(\widehat{\sigma}^2/2)\). \], \[
E.g., if you fit a model y ~ log(x1) + log(x2), and transform is True, then you can pass a data structure that contains x1 and x2 in their original form. \left[ \exp\left(\widehat{\log(Y)} - t_c \cdot \text{se}(\widetilde{e}_i) \right);\quad \exp\left(\widehat{\log(Y)} + t_c \cdot \text{se}(\widetilde{e}_i) \right)\right]
Ie., we do not want any expansion magic from using **2, Now we only have to pass the single variable and we get the transformed right-hand side variables automatically. For larger samples sizes \(\widehat{Y}_{c}\) is closer to the true mean than \(\widehat{Y}\). \]
\begin{aligned}
Prediction vs Forecasting¶ The results objects also contain two methods that all for both in-sample fitted values and out-of-sample forecasting. \end{aligned}
Interpreting the Prediction Interval. \log(Y) = \beta_0 + \beta_1 X + \epsilon
&= \mathbb{C}{\rm ov} (\widetilde{\boldsymbol{\varepsilon}}, \widetilde{\mathbf{X}} \left( \mathbf{X}^\top \mathbf{X}\right)^{-1} \mathbf{X}^\top \mathbf{Y})\\
\mathbb{E} \left[ (Y - \mathbb{E} [Y|\mathbf{X}])^2 \right] = \mathbb{E}\left[ \mathbb{V}{\rm ar} (Y | X) \right]. \[
Thus, \(g(\mathbf{X}) = \mathbb{E} [Y|\mathbf{X}]\) is the best predictor of \(Y\). statsmodels logistic regression predict, Simple logistic regression using statsmodels (formula version) Linear regression with the Associated Press # In this piece from the Associated Press , Nicky Forster combines from the US Census Bureau and the CDC to see how life expectancy is related to actors like unemployment, income, and others. \], \[
Assume that the data really are randomly sampled from a Gaussian distribution. &= \sigma^2 \left( \mathbf{I} + \widetilde{\mathbf{X}} \left( \mathbf{X}^\top \mathbf{X}\right)^{-1} \widetilde{\mathbf{X}}^\top\right)
\], \[
\], \(\widetilde{\mathbf{X}} \boldsymbol{\beta}\), \[
\begin{aligned}
&= \mathbb{V}{\rm ar}\left( \widetilde{\mathbf{Y}} \right) - \mathbb{C}{\rm ov} (\widetilde{\mathbf{Y}}, \widehat{\mathbf{Y}}) - \mathbb{C}{\rm ov} ( \widehat{\mathbf{Y}}, \widetilde{\mathbf{Y}})+ \mathbb{V}{\rm ar}\left( \widehat{\mathbf{Y}} \right) \\
Home; Uncategorized; statsmodels ols multiple regression; statsmodels ols multiple regression [10.83615884 10.70172168 10.47272445 10.18596293 9.88987328 9.63267325 9.45055669 9.35883215 9.34817472 9.38690914] \]
The prediction interval around yhat can be calculated as follows: 1. yhat +/- z * sigma. # Let's calculate the mean resposne (i.e. Let's utilize the statsmodels package to streamline this process and examine some more tendencies of interval estimates.. Let \(\widetilde{X}\) be a given value of the explanatory variable. Using formulas can make both estimation and prediction a lot easier, We use the I to indicate use of the Identity transform. \mathbb{E} \left[ (Y - g(\mathbf{X}))^2 \right] &= \mathbb{E} \left[ (Y + \mathbb{E} [Y|\mathbf{X}] - \mathbb{E} [Y|\mathbf{X}] - g(\mathbf{X}))^2 \right] \\
import statsmodels.stats.proportion as smp # e.g. \end{aligned}
There is a 95 per cent probability that the real value of y in the population for a given value of x lies within the prediction interval. OLS method. Our second model also has an R-squared of 65.76%, but again this doesn’t tell us anything about how precise our prediction interval will be. A prediction interval relates to a realization (which has not yet been observed, but will be observed in the future), whereas a confidence interval pertains to a parameter (which is in principle not observable, e.g., the population mean). Assume that the data really are randomly sampled from a Gaussian distribution. \[
\widehat{Y}_i \pm t_{(1 - \alpha/2, N-2)} \cdot \text{se}(\widetilde{e}_i)
3.7 OLS Prediction and Prediction Intervals, Hence, a prediction interval will be wider than a confidence interval. \]
Prediction plays an important role in financial analysis (forecasting sales, revenue, etc. or more compactly, \(\left[ \exp\left(\widehat{\log(Y)} \pm t_c \cdot \text{se}(\widetilde{e}_i) \right)\right]\). In this exercise, we've generated a binomial sample of the number of heads in 50 fair coin flips saved as the heads variable. \]
; transform (bool, optional) – If the model was fit via a formula, do you want to pass exog through the formula.Default is True. \[
\[
We have examined model specification, parameter estimation and interpretation techniques. \]. &= 0
Y = \exp(\beta_0 + \beta_1 X + \epsilon)
&= \mathbb{E}(Y|X)\cdot \exp(\epsilon)
In order to do so, we apply the same technique that we did for the point predictor - we estimate the prediction intervals for \(\widehat{\log(Y)}\) and take their exponent. ie., The default alpha = .05 returns a 95% confidence interval. \mathbb{E} \left[ (Y - g(\mathbf{X}))^2 \right] &= \mathbb{E} \left[ (Y + \mathbb{E} [Y|\mathbf{X}] - \mathbb{E} [Y|\mathbf{X}] - g(\mathbf{X}))^2 \right] \\
&= \mathbb{E}(Y|X)\cdot \exp(\epsilon)
If you sample the data many times, and calculate a confidence interval of the mean from each sample, youâd expect about \(95\%\) of those intervals to include the true value of the population mean. \[
Prediction intervals tell you where you can expect to see the next data point sampled. \[
\]
\text{argmin}_{g(\mathbf{X})} \mathbb{E} \left[ (Y - g(\mathbf{X}))^2 \right]. \[
Statsmodels is a Python module that provides classes and functions for the estimation of ... prediction interval for a new instance. Then sample one more value from the population. In our case: There is a slight difference between the corrected and the natural predictor when the variance of the sample, \(Y\), increases. The sm.OLS method takes two array-like objects a and b as input. from IPython.display import HTML, display import statsmodels.api as sm from statsmodels.formula.api import ols from statsmodels.sandbox.regression.predstd import wls_prediction_std import matplotlib.pyplot as plt import seaborn as sns %matplotlib inline sns.set_style("darkgrid") import pandas as pd import numpy as np Taking \(g(\mathbf{X}) = \mathbb{E} [Y|\mathbf{X}]\) minimizes the above equality to the expectation of the conditional variance of \(Y\) given \(\mathbf{X}\):
Calculate and plot Statsmodels OLS and WLS confidence intervals - ci.py. the prediction is comprised of the systematic and the random components, but they are multiplicative, rather than additive. \]
&= \mathbb{E}\left[ \mathbb{V}{\rm ar} (Y | X) \right] + \mathbb{E} \left[ (\mathbb{E} [Y|\mathbf{X}] - g(\mathbf{X}))^2\right]. \]
&= \mathbb{E} \left[ (Y - \mathbb{E} [Y|\mathbf{X}])^2 + 2(Y - \mathbb{E} [Y|\mathbf{X}])(\mathbb{E} [Y|\mathbf{X}] - g(\mathbf{X})) + (\mathbb{E} [Y|\mathbf{X}] - g(\mathbf{X}))^2 \right] \\
\begin{aligned}
\], \(\mathbb{E}\left(\widetilde{Y} | \widetilde{X} \right) = \beta_0 + \beta_1 \widetilde{X}\), \[
\widehat{Y}_{c} = \widehat{\mathbb{E}}(Y|X) \cdot \exp(\widehat{\sigma}^2/2) = \widehat{Y}\cdot \exp(\widehat{\sigma}^2/2)
The difference from the mean response is that when we are talking about the prediction, our regression outcome is composed of two parts:
fitted) values again: # Prediction intervals for the predicted Y: #from statsmodels.stats.outliers_influence import summary_table, #dt = summary_table(lm_fit, alpha = 0.05)[1], #yprd_ci_lower, yprd_ci_upper = dt[:, 6:8].T, \(\mathbb{E} (\boldsymbol{Y}|\boldsymbol{X})\), \(\widehat{\mathbf{Y}} = \mathbb{E} (\boldsymbol{Y}|\boldsymbol{X})\), \(\widetilde{\mathbf{X}} \widehat{\boldsymbol{\beta}}\), \[
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For \ ( \widehat { Y } \ ) that the second model an...