Time Series Analysis Quiz

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Questions and Answers

What does transformation of the forecast variable aim to achieve when dealing with seasonal variation?

Ensure the seasonal swings remain constant at all levels.

Increase the accuracy of all forecast models.

Produce a time series with constant seasonal variation. (correct)

Reduce the impact of outliers on the model.

Which of the following is NOT a type of seasonal variation?

Regular seasonal variation

Decreasing seasonal variation (correct)

Constant seasonal variation

Increasing seasonal variation

When applying a natural logarithm transformation, which of the following forms is used?

y* = ln(yt) (correct)

y* = yt^0.5

y* = yt^2

y* = yt^3

In the equation y* = yt^λ, what does the parameter λ represent when transforming data?

The power to which the variable is raised. Signup and view all the answers

Which of the following transformations can help when the time series exhibits increasing seasonal variation?

Square root transformation. Signup and view all the answers

What does a polynomial trend indicate in time series regression?

One or more reversals in curvature. Signup and view all the answers

What does the error term (εt) in the time series regression model represent?

A random fluctuation and deviation from the average level. Signup and view all the answers

Which of the following is true about a time series with a linear trend?

Its slope remains constant over time. Signup and view all the answers

In time series regression, what would a model with no trend signify?

The absence of long-run growth or decline. Signup and view all the answers

Which equation represents a quadratic trend model in time series regression?

yt = β0 + β1t + β2t^2 Signup and view all the answers

What assumption is made about the error term (εt) in time series regression?

It must be independent and satisfy constant variance. Signup and view all the answers

Which of the following describes a p-th order polynomial trend?

Indicates multiple changes in the direction of the data. Signup and view all the answers

Why might a company want to forecast its monthly cod catch using time series regression?

To plan the operations of its fish processing plant effectively. Signup and view all the answers

What does the Durbin-Watson statistic primarily assess in a regression analysis?

The independence of residuals Signup and view all the answers

For a sample size of 10 and two predictors, what is the upper critical value (dU) of the Durbin-Watson statistic at the 0.05 significance level?

1.70 Signup and view all the answers

What indicates potential heteroscedasticity in residuals when analyzing residual vs fitted plots?

A discernible pattern in the spread of points Signup and view all the answers

At a significance level of 0.05, what is the lower critical value (dL) for a sample size of 15 and three predictors?

0.90 Signup and view all the answers

How many predictors are assessed if the Durbin-Watson critical values are listed up to k = 4?

4 Signup and view all the answers

Which of the following scenarios corresponds to a Durbin-Watson statistic value below dL?

Positive autocorrelation Signup and view all the answers

If a student calculates a Durbin-Watson statistic of 0.75 with 12 observations and 2 predictors, what does this suggest?

The model has positive autocorrelation Signup and view all the answers

For a regression analysis, which condition must be met regarding the variance of residuals for the Durbin-Watson test to be valid?

Residuals must have constant variance Signup and view all the answers

What can be concluded if the Durbin-Watson statistic falls between dL and dU?

Decision is inconclusive Signup and view all the answers

What is the null hypothesis (H0) for testing negative autocorrelation in error terms?

The error terms are not autocorrelated Signup and view all the answers

In a Durbin-Watson test for autocorrelation, which of the following residual patterns would indicate acceptable conditions?

No apparent pattern in residuals Signup and view all the answers

In testing for negative autocorrelation, what condition leads to rejecting the null hypothesis?

If (4 - d) is less than dL,α Signup and view all the answers

For the Durbin-Watson test, if d < dL,α or (4 - d) < dL,α, what is the conclusion?

Reject H0 Signup and view all the answers

What does the Durbin-Watson statistic measure?

The degree of autocorrelation in residuals Signup and view all the answers

Given dL,0.05=1.27 and dU,0.05=1.45, what does a Durbin-Watson statistic of 1.682 indicate?

No evidence of autocorrelation Signup and view all the answers

If the Durbin-Watson test statistic falls within the range dL,α ≤ d ≤ dU,α, what does this imply?

The test is inclusive Signup and view all the answers

What is the alternative hypothesis (H1) when testing for both positive and negative autocorrelation?

The error terms are positively or negatively autocorrelated Signup and view all the answers

In the example described, what equation represents the prediction model for the calculator sales?

yˆt = 198.02899 + 8.07435t Signup and view all the answers

What is the purpose of Smith's inventory policy for the Bismark X-12 calculators?

To balance demand fulfillment and minimize excess inventory Signup and view all the answers

Based on the information, what does the regression model aim to forecast?

Future sales of the Bismark X-12 calculators Signup and view all the answers

Which variable represents the slope in the regression formula used to predict sales?

β1 Signup and view all the answers

How is the point forecast for future sales calculated?

Using the linear regression equation derived from historical data Signup and view all the answers

What does the term 'prediction interval' represent in the forecasting process?

The range within which the forecasted sales are expected to fall Signup and view all the answers

What is the estimated value of β1, the slope of the regression line?

8.07435 Signup and view all the answers

What does the variable β0 represent in the regression model?

The initial sales value when time is zero Signup and view all the answers

Which of the following factors is NOT considered in Smith's forecasting method?

Current stock levels Signup and view all the answers

What calculation is necessary to obtain a prediction interval for future sales?

Variances and t-values from sample data Signup and view all the answers

What is one implication of a trend that increases linearly over time in the context of sales forecasting?

Sales are likely to continue rising if past trends hold Signup and view all the answers

What does the model suggest when β1 > 1 in the growth model?

The data exhibits exponential growth. Signup and view all the answers

What are the least point estimates of α0 and α1?

α0 = 2.07012, α1 = 0.25688 Signup and view all the answers

How is the point estimate for the growth rate calculated?

100 * (β1 − 1) Signup and view all the answers

What is the correct interpretation of the prediction interval for y16?

It provides a range within which the actual value of y16 is expected to fall. Signup and view all the answers

What hypothesis is being tested with the Durbin-Watson statistic?

The error terms are independent or not autocorrelated. Signup and view all the answers

What is the estimate for y16 derived from the model?

483.09 Signup and view all the answers

What does an adjusted R2 value indicate when selecting models?

Higher values suggest a better fit of the model to the data. Signup and view all the answers

What is the significance of using Cross-Validation in model selection?

It ensures lower MSE is obtained from testing data. Signup and view all the answers

Study Notes

Time Series Regression

A model that relates the dependent variable (yt) to functions of time.
Used when parameters describing the time series remain constant over time.
- This means that if a time series has a linear trend, the slope of the trend line remains constant.
- If a time series has a monthly seasonal component, the seasonal parameter for each month stays the same from year to year.

Modeling Trend Using Polynomial Function

A time series (yt) can be described using a trend model, written as:
- yt = TRt + εt
- where:
  - yt = the value of the time series in period t
  - TRt = the trend in time period t
  - εt = the error term in time period t
The time series can be represented by an average level (μt = TRt) and an error term (εt).
The error term represents random fluctuations from the deviation between yt values and the average level (μt).

Some Common Trends

No trend: Implies no long-run growth or decline in the time series. (TRt = β₀)
Linear trend: Implies a straight-line long-run growth or decline. (TRt = β₀ + β₁t)
Quadratic trend: Implies a quadratic long-run change (growth or decline with an increasing or decreasing rate). (TRt = β₀ + β₁t + β₂t²)

p-th Order Polynomial Trend

Indicates one or more reversals in curvature.
TRt = β₀ + β₁t + β₂t² + ... + βptp
Parameters can be estimated using regression techniques (e.g., least squares method).
The error term (εt) is assumed to have constant variance, independence, and normality.

Example 6.1 (Bay City Seafood Company)

The company wants to forecast monthly cod catch (in tons).
Data from the past two years (years 1 and 2) show the cod catch fluctuating around a constant average level.
A regression model is used to forecast future cod catch. The point estimate for β₀ (the average cod catch) is 351.29.

Least Squares Point Estimate of β₀

β₀ = (Σyt) / T, where Σyt is the sum of all yt values, and T is the number of periods.

100(1 – α)% Prediction Interval for yt

ӯt ± tα/2s√(1 + 1/T), where ӯt is the point prediction of yt, t[α/2] is the critical value from the t-distribution, and s is the standard error of the regression.

Other Examples and Concepts

The note contains numerous examples, including those involving different types of data (e.g., calculator sales, loan requests), plots of these data over time, regression models, and code snippets illustrative of the process.
Covers polynomial, trigonometric and growth curve models.
Discusses the modeling of seasonal variation using dummy variables.
Contains relevant formal tests such as the Durbin-Watson test for autocorrelation and an explanation of its use.
Explores issues such as the appropriate choice of a model for forecasting (e.g., when to use different transformations of the data to remove issues like increasing seasonal variation).
Discusses model evaluation, and aspects of choosing the correct predictor for a model.

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Description

Test your knowledge on key concepts in time series analysis, including transformations, seasonal variation, and regression models. This quiz covers important aspects such as polynomial trends, error terms, and forecasting techniques. Perfect for students and professionals delving into statistical methods.