(SEM VII) THEORY EXAMINATION 2023-24 TIME SERIES ANALYSIS AND FORECASTING

SECTION A – Very Short Answer Type

(2 × 10 = 20 Marks)

a) Different types of data used in time series analysis

The main types are time series data (observations over time), cross-sectional data (data at one point in time), and panel data (combination of time series and cross-sectional data).

b) Autocorrelation

Autocorrelation measures the correlation between current and past values of a time series. It helps identify dependence across time lags.

c) Metrics to evaluate forecasting performance

Common metrics include MAE (Mean Absolute Error), MSE (Mean Squared Error), RMSE, and MAPE, which measure forecast accuracy.

d) Purpose of plotting smoothed data

Smoothed data helps remove random noise and clearly reveal trend and seasonal patterns in a time series.

e) Least squares estimation

Least squares estimation finds model parameters by minimizing the sum of squared errors between observed and predicted values.

f) Variable selection methods in regression

Methods include forward selection, backward elimination, and stepwise regression, used to choose the most relevant variables.

g) Identifying order of ARIMA model

The order is identified using ACF and PACF plots along with differencing tests for stationarity.

h) Stationarity in ARMA models

A time series is stationary if its mean, variance, and autocovariance remain constant over time.

i) Seasonal ARIMA model

Seasonal ARIMA (SARIMA) extends ARIMA by including seasonal autoregressive and moving average terms.

j) Vector AR models

Vector Autoregressive (VAR) models capture interdependencies among multiple time series by modeling each variable using past values of all variables.

SECTION B – Long Answer Type

(Attempt any three – 10 Marks each)

2(a) Time Series and Its Components

A time series is a sequence of observations recorded at regular time intervals.

Components:

Trend – Long-term movement (e.g., increasing sales)

Seasonality – Regular periodic fluctuations (e.g., festival sales)

Cyclic changes – Long-term economic cycles

Irregular component – Random, unpredictable variations

These components help analysts understand patterns and improve forecasting accuracy.

2(b) Standard Approach to Time Series Forecasting

Steps include:                                                           Graphical analysis

Data cleaning and transformation                           Stationarity checking

Model identification (AR, MA, ARIMA)                     Parameter estimation

Diagnostic checking                                                 Forecast generation

This structured approach ensures reliable forecasts.

2(c) Model Adequacy Checking in Linear Regression

Adequacy checking ensures that assumptions are satisfied:

Residual analysis                                                       Normality tests

Homoscedasticity tests                                             Autocorrelation checks

If violations exist, transformations or alternative models are applied.

2(d) Strengths and Limitations of ARIMA Forecasts

Strengths:                                                                   Effective for short-term forecasting

Flexible and data-driven

Limitations:                                                                 Requires stationary data

Weak for long-term forecasts                                    Does not handle external variables directly

2(e) Predicting with Seasonal ARIMA

SARIMA incorporates seasonal patterns using past seasonal data.

Examples:                                                                 Monthly electricity consumption

Quarterly sales                                                           Daily temperature patterns

It improves forecasts where seasonality is strong.

SECTION C – Descriptive Answer Type

3(a) Nature and Uses of Forecasting

Forecasting predicts future values based on historical data.

Uses:                                                                           Business planning

Demand forecasting                                                   Weather prediction

Stock market analysis                                                 Inventory control

Forecasting supports better decision-making and risk management.

3(b) Forecasting Process

Steps include:                                                             Data collection

Data preprocessing                                                    Model selection

Model estimation                                                       Validation

Forecast output

Resources required include historical data, computational tools, and domain expertise.

4(a) General Framework for Time Series Forecasting

The framework includes:                                           Exploratory data analysis

Stationarity testing                                                    Model identification

Parameter estimation                                                Diagnostic checking

Forecast generation

This ensures a systematic forecasting approach.

4(b) Need for Data Transformations

Transformations such as differencing, inflation adjustment, and holiday correction are applied to:

Remove trends                                                           Stabilize variance

Improve stationarity

Example: Differencing removes upward sales trends.

5(a) Exponential Smoothing Models

First-order (Simple) smoothing:                                Used for data without trend or seasonality.

Second-order smoothing:                                         Handles trends using two smoothing equations.

These models are simple and efficient for short-term forecasts.

5(b) Generalized and Weighted Least Squares

GLS and WLS address heteroscedasticity and autocorrelation, unlike ordinary least squares.

They assign different weights to observations, improving estimation accuracy.

6(a) AR and MA Models

AR models use past values                                       MA models use past errors

ARMA combines both

They capture serial correlation patterns effectively.

6(b) Role of ACF and PACF

ACF identifies MA terms                                           PACF identifies AR terms

They help in model selection and adequacy checking.

7(a) Multivariate Time Series Models

Multivariate models analyze interrelated time series simultaneously.

Advantages:                                                              Captures interactions

Improves forecasting accuracy                                 Univariate models analyze variables independently.

7(b) Model Selection Criteria

Common criteria include:

AIC                                                                            BIC

MAPE                                                                        RMSE

Lower values indicate better model performance.