How much of the variation in is explained by the model? (Write the answer in decimal form. Keep 2 decimal points.)

Question

How much of the variation in  is explained by the model? (Write the answer in decimal form. Keep 2 decimal points.)

BlackTom AI · Accepted Answer

This item asks for the proportion of variation explained by a model, expressed as a decimal with two decimal places. However, the prompt as provided is missing key context, notably: the dependent variable being analyzed and the data or model output needed to compute the variation explained. Because of that, we cannot produce a numeric answer yet.

First, recall the standard metric used to quantify how much of the variation in the outcome is explained by a model: R-squared (R^2). It is computed from sums of squares in regression analysis. The steps are:
- Identify the observed values of the dependent variable, y_i, and the predicted values from the model, ŷ_i.
- Compute the total sum of squares (SS_tot) as the sum over all observations of (y_i − ȳ)^2, where ȳ is the mean of the observed values.
- Compute the residual sum of squares (SS_res) as the sum over all observations of (y_i − ŷ_i)^2.
- Calculate R^2 = 1 − (SS_res / SS_tot). This value lies between 0 and 1 and represents the fraction of variability in y explained by the model.

If the prompt expects two decimal places, you would round R^2 to two decimals. For example, R^2 = 0.8765 would be reported as 0.88, and R^2 = 0.1234 as 0.12.

Notes and nuances:
- If the model includes more predictors, R^2 can be high simply due to overfitting. In such cases, adjusted R^2 provides a penalty for adding predictors that do not improve fit substantively.
- If there is a mistake in data entry or model specification, SS_res could be equal to SS_tot, yielding R^2 ≈ 0, or even negative when using certain algorithmic contexts, though conventional ordinary least squares regression yields R^2 in [0,1].
- The instruction mentions “variation in [outcome] explained by the model,” which reinforces that R^2 is the appropriate metric, rather than other metrics like RMSE or MAE which quantify error rather than explained variance.

Because the necessary data (the actual y values and the model predictions) are not provided in the prompt, you cannot compute the numeric R^2 value here. Once you have y_i and ŷ_i (or the model’s summary output that provides SS_res and SS_tot), apply the formula above and format the result to two decimals as requested.

ECON 4818-581 Final Exam

How much of the variation in is explained by the model? (Write the answer in decimal form. Keep 2 decimal points.)

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