Exam-style answers for every QP question.

Theory answer

Forecasting is the process of estimating future values of a variable by using past data, present information, and suitable assumptions. In time series analysis, forecasting helps in planning production, inventory, finance, staffing, sales targets, and policy decisions.

1. Qualitative forecasting methods are judgement-based methods. They are used when past numerical data is limited, unreliable, or not directly relevant. Examples include expert opinion, Delphi method, market survey, consumer intention survey, sales-force composite method, and historical analogy. These methods are useful for new products, new markets, or long-term strategic decisions, but they may be subjective.
2. Quantitative forecasting methods are data-based methods. They use numerical past observations and mathematical models. Examples include moving averages, weighted moving averages, exponential smoothing, decomposition, regression, and ARIMA models. These methods are objective and reproducible, but they require reliable historical data and may fail if the future pattern changes suddenly.
3. Difference: qualitative methods depend mainly on experience and judgement, whereas quantitative methods depend mainly on recorded data and statistical calculations.

Theory answer

A time series is a set of observations recorded in time order. The observed value is usually affected by four components.

1. Trend component: The long-term upward or downward movement in the data. It shows the general direction over a long period, such as increasing sales or falling demand.
2. Seasonal component: A regular pattern that repeats within a fixed short period such as month, quarter, week, or day. Examples are festival sales, summer electricity demand, and quarterly demand patterns.
3. Cyclical component: Wave-like movements around the trend over periods longer than one year. These are often connected with business cycles such as boom, recession, depression, and recovery.
4. Irregular component: Random, accidental, or unpredictable variation caused by events such as strikes, floods, pandemics, sudden policy changes, or one-time market shocks.

The common models are additive model Y = T + S + C + I and multiplicative model Y = T x S x C x I. Additive model is preferred when seasonal variation is roughly constant; multiplicative model is preferred when seasonal variation changes in proportion to the level of the series.

Theory answer

The trend is the long-term movement of a time series. The main methods used to find trend are:

1. Graphic or freehand curve method: Plot the data on a graph and draw a smooth line through the points by judgement. It is simple, but subjective.
2. Semi-average method: Divide the data into two equal parts, calculate the average of each part, plot these averages at their middle time points, and join them. It is easy, but it may be affected by unusual values.
3. Moving average method: Calculate averages of fixed-size consecutive groups. It smooths short-term fluctuations and gives a clearer trend. It is useful when trend is not strictly linear.
4. Method of least squares: Fit a mathematical trend equation, such as a straight line y = a + bx or parabola y = a + bx + cx^2, by minimizing the sum of squared errors. It is objective and suitable for forecasting.

Theory answer

Decomposition is the process of separating a time series into its major components: trend, seasonal, cyclical, and irregular variations. This helps us understand the structure of the data and produce better forecasts.

1. Choose the model. In additive decomposition, Y = T + S + C + I. In multiplicative decomposition, Y = T x S x C x I.
2. Estimate the level or trend of the series. For this two-year monthly question, the yearly average is used as the level for each year.
3. Estimate seasonal effects. For additive decomposition, subtract the yearly average from each month. For multiplicative decomposition, divide each monthly value by the yearly average.
4. Average the same months across years to obtain the seasonal index for each month.
5. Adjust the seasonal indices. Additive seasonal indices should sum to zero. Multiplicative seasonal indices should average to one.

Theory answer

In decomposition, the observed value is broken into systematic parts so that the seasonal pattern and trend pattern can be studied separately. The steps are: identify the model, estimate the annual or trend level, remove the level from each monthly value, average the same months, and adjust the indices.

Additive decomposition is used when seasonal changes are measured in the same units as the data. Multiplicative decomposition is used when seasonal changes are proportional to the level of the series.

Theory answer

Decomposition separates the original time series into long-term movement and repeated seasonal effects. The additive form is Y = T + S + C + I, and the multiplicative form is Y = T x S x C x I. After seasonal indices are found, they can be used to seasonally adjust data or to prepare seasonal forecasts.

Theory answer

The purpose of decomposition is to make a time series easier to understand by separating trend and seasonality. The analyst first estimates the base level, then measures how each month differs from that base. The repeated monthly pattern becomes the seasonal component.

For additive decomposition, seasonal effects are added to the level. For multiplicative decomposition, seasonal factors multiply the level. A multiplicative model is often suitable when the second year has a higher level and seasonal lifts also scale upward.

Theory answer

1. Single exponential smoothing: Used when data has level but no clear trend or seasonality. It updates the forecast using the latest forecast error.
2. Double exponential smoothing or Holt's method: Used when data has a trend. It updates both level and trend, so it can forecast increasing or decreasing series.
3. Triple exponential smoothing or Holt-Winters method: Used when data has both trend and seasonality. It may be additive or multiplicative depending on the seasonal pattern.
4. Damped trend smoothing: A variation where the trend effect is gradually reduced for longer forecast horizons.

Theory answer

Exponential smoothing methods give higher weight to recent observations and lower weight to older observations. The smoothing parameter alpha lies between 0 and 1. A small alpha gives a smoother forecast, while a large alpha reacts faster to recent changes.

The main types are single exponential smoothing for level-only data, double exponential smoothing for trend data, and triple exponential smoothing for trend plus seasonal data.

Theory answer

Exponential smoothing methods are recursive forecasting methods. They are easy to update because each new forecast is obtained from the previous smoothed values and the latest actual observation.

1. SES handles only a changing level.
2. DES handles level and trend using alpha for level smoothing and beta for trend smoothing.
3. Holt-Winters handles level, trend, and seasonality.

Theory answer

Single exponential smoothing is written as F_(t+1) = alpha y_t + (1 - alpha)F_t. It is best for data without a strong trend. A larger alpha places more weight on the latest observation and gives a more responsive forecast.

Double exponential smoothing adds a trend component, and triple exponential smoothing adds both trend and seasonal components.

Theory answer

Exponential smoothing forecasts are weighted averages where recent observations receive more importance. The three core types are SES, DES, and Holt-Winters. DES is appropriate here because the question asks us to estimate both level and trend through Holt's recurrence equations.

Theory answer

The types of exponential smoothing are selected according to the pattern in the data. SES is used for level-only series, DES is used when the series has trend, and Holt-Winters is used when there is trend plus seasonality. Since this sales series has a general upward direction, double exponential smoothing is suitable.