f T denotes the number of observations, which of the following equations represents the one-step-ahead forecast for the model yt = β0 + β1xt + εt?

Multiple Choice

yˆT+1 = b0 + b1xT+1

yˆT+1 = b0 − b1xT+1

yˆT+1 = b0 + b1xT + 1

yˆT+1 = b0 + b1xT + 1

In the context of a simple linear regression model, the one-step-ahead forecast for \( y_t \) is given by:

\[ y_t = \beta_0 + \beta_1 x_t + \epsilon_t \]

Where:
- \( \beta_0 \) is the intercept
- \( \beta_1 \) is the slope coefficient
- \( x_t \) is the independent variable at time \( t \)
- \( \epsilon_t \) is the error term

For a one-step-ahead forecast, we predict the value of \( y \) at time \( T+1 \) based on the value of the independent variable at time \( T+1 \). Thus, the forecasted value \( \hat{y}_{T+1} \) would be:

\[ \hat{y}_{T+1} = \beta_0 + \beta_1 x_{T+1} \]

Given the multiple-choice options, the correct equation that represents the one-step-ahead forecast is:

\[ \hat{y}_{T+1} = b_0 + b_1 x_{T+1} \]

where \( b_0 \) and \( b_1 \) are the estimated coefficients from the regression model.

So, the correct answer is:

\[ \hat{y}_{T+1} = b_0 + b_1 x_{T+1} \]

Do you want details or have any questions?

Here are some relative questions for further exploration:
1. How are the coefficients \( \beta_0 \) and \( \beta_1 \) estimated in a simple linear regression model?
2. What assumptions are made about the error term \( \epsilon_t \) in linear regression?
3. How does one assess the goodness of fit for a linear regression model?
4. What is the difference between a point forecast and an interval forecast?
5. How would you extend this model to include multiple predictors?
6. What is autocorrelation, and why is it important in time series forecasting?
7. How do you handle seasonality in time series forecasting models?
8. What are some common pitfalls when interpreting the results of a regression analysis?

**Tip:** Always check the assumptions of your regression model (linearity, independence, homoscedasticity, normality) to ensure the validity of your results.

Understand the one-step-ahead forecast in a linear regression model with this detailed explanation. Learn how to predict future values using regression coefficients and independent variables. Suitable for advanced undergraduate students and anyone interested in statistical forecasting.

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