CAIIB ABM UNIT 5 MCQs – Time Series

Correct Answer: C. It comprises observations gathered over successive points or periods in time. A time series is defined by the sequential collection of data points at regular intervals or successive moments.

Correct Answer: C. There are 4 types of variations in time series. 1.Secular Trend, 2.Cyclical Fluctuation, 3.Seasonal Variation and 4.Irregular Variation

Correct Answer: B. The smooth, long-term direction of the data over an extended period. Secular trend captures the general tendency of the data, whether it is increasing, decreasing, or remaining constant over a long time frame.

Correct Answer: D. They occur over periods longer than a year with varying durations and amplitudes. Cyclical fluctuations, such as business cycles, extend beyond one year and do not follow a regular pattern in terms of timing or intensity.

Correct Answer: C. Complete a regular cycle within one year or less. Seasonal variations are characterised by predictable patterns of change that repeat annually or within shorter intervals like quarters or months.

Correct Answer: D. Irregular Variation. Irregular variations are caused by random, unpredictable events like natural disasters or wars, whose effects on the time series cannot be foreseen.

Correct Answer: C. To project past long-term patterns into the future for forecasting. Studying secular trends helps understand the historical long-term direction of data, which can then be extrapolated to predict future values. For example, if a company’s sales have shown a consistent average increase of 10% per year over the last decade (the trend), this trend can be used to estimate sales for the next few years, assuming underlying conditions remain similar.

Correct Answer: B. A linear trend is represented by a straight line, while a curvilinear trend is represented by a curve. A linear trend implies a constant amount or rate of change (y = a + bx), like a factory consistently increasing output by 100 units each year. A curvilinear trend involves a changing rate of growth or decline (y = a + bx + cx^2), such as the sales of a new product which might grow slowly initially, then rapidly, and finally level off.

Correct Answer: C. 2. For an odd number of years, the mean year is the middle year (2020). The coded value for a year ‘T’ is calculated as x = T - mean_year. Therefore, for 2022, the coded value is x = 2022 - 2020 = 2.

Correct Answer: C. -5. For an even number of years, the mean time falls between the two middle years (between 2019 and 2020, i.e., 2019.5). The initial codes (T - mean_time) would be -2.5, -1.5, -0.5, 0.5, 1.5, 2.5. To avoid fractions, these are multiplied by 2. Thus, the coded value for 2017 becomes x = 2 * (2017 - 2019.5) = 2 * (-2.5) = -5.

Correct Answer: B. 7.5. When the time variable ‘x’ is coded such that its mean is 0 (making Σx = 0), the slope ‘b’ is calculated using the formula b = Σxy / Σx². Substituting the given values: b = 1200 / 160 = 7.5.

Correct Answer: B. 140. When the coded time ‘x’ has a mean of 0, the y-intercept ‘a’ is equal to the mean of the dependent variable ‘y’ (ȳ). The mean ȳ is calculated as ȳ = Σy / n, where n is the number of observations. Here, ȳ = 1120 / 8 = 140. Therefore, a = 140.

Correct Answer: B. Cyclical variation tends to oscillate above and below the secular trend line for periods longer than a year, while seasonal variation makes a complete regular cycle within each year. Cyclical variation, often associated with business cycles, extends over multiple years, showing expansions and contractions around the long-term trend. Seasonal variation, however, completes a consistent pattern within a single year, such as increased sales during festive seasons or higher utility consumption in specific weather conditions.

Correct Answer: B. To remove the effects of seasonal variation from a time series to better analyse other components. Deseasonalisation involves adjusting time series data to eliminate the regular, within-a-year patterns caused by seasonality. This process helps in isolating and understanding the underlying trend and cyclical components, which might be obscured by seasonal fluctuations. For example, removing the effect of higher retail sales during the holiday season allows for a clearer view of the overall growth trend in sales.

Correct Answer: C. Dividing the actual observed data by the trend and predicted data and multiplying by 100. The Residual Method, when applied to isolate cyclical variation after identifying the secular trend, calculates the cyclical component as a percentage of the trend. This is typically done by the formula (Actual Value ⁄ Trend Value) ∗ 100, showing how the actual data deviates from the estimated trend due to cyclical and irregular factors. Another measure mentioned is the Relative Cycle Residual, calculated as {(Actual Value − Predicted Value) ⁄ Predicted Value} ∗ 100.

Correct Answer: B. (Actual Value ⁄ Predicted Value) ∗ 100. The percentage of trend is calculated by dividing the actual time series value ($y$) by the predicted value ($\hat{y}$) obtained from the trend line and multiplying the result by 100. This gives an indication of how much the actual value is as a percentage of the trend value, with deviations from 100% indicating the presence of cyclical and irregular components.

Correct Answer: B. {(Actual Value−Predicted Value)⁄(Predicted Value)}∗100. The Relative Cycle Residual measures the percentage deviation from the trend for each value. It is calculated by finding the difference between the actual time series value and the predicted trend value, dividing this difference by the predicted trend value, and then multiplying by 100.

Correct Answer: A. 98.7%. The percentage of trend is calculated as (Actual Value ⁄ Estimated Trend Value) ∗ 100. In this case, (75 ⁄ 76) ∗ 100 ≈ 98.7%. This indicates that the actual value was approximately 98.7% of the estimated trend value for that period.

Correct Answer: B. -1.3%. The Relative Cyclical Residual is calculated as {(Actual Value−Estimated Trend Value)⁄(Estimated Trend Value)}∗100. In this case, {(75−76)⁄76}∗100 = (−1⁄76)∗100 ≈ −1.3. This means the actual value was 1.3% below the estimated trend value.

Correct Answer: B. 102.5%. The percentage of trend is calculated as (Actual Value ⁄ Estimated Trend Value) ∗ 100. Here, (82 ⁄ 80) ∗ 100 = 1.025 ∗ 100 = 102.5%. This indicates the actual value is 102.5% of the estimated trend value.

Correct Answer: A. 2.5%. The Relative Cyclical Residual is calculated as {(Actual Value−Estimated Trend Value)⁄(Estimated Trend Value)}∗100. Here, {(82−80)⁄80}∗100 = (2⁄80)∗100 = 0.025∗100 = 2.5. This shows the actual value was 2.5% above the estimated trend value.

Correct Answer: C. 1930 rooms. To obtain the estimated occupancy considering seasonality, the deseasonalised estimate is multiplied by the seasonal index (expressed as a fraction of 100). The calculation is: Deseasonalised Estimate ∗ (Seasonal Index ⁄ 100) = 2121 ∗ (91.0 ⁄ 100) = 2121 ∗ 0.91 = 1929.1. Rounded to the nearest whole number as occupancy is in whole rooms, this is 1930 rooms.

Correct Answer: C. They are discarded to eliminate extreme cyclical and irregular components. The modified mean of seasonal indices is calculated by excluding the highest and lowest values for each specific season (e.g., each quarter). This process helps to reduce the impact of unusually large or small fluctuations caused by cyclical or irregular factors, resulting in a more stable and representative measure of the typical seasonal pattern.

Correct Answer: B. When the number of time intervals within a year (e.g., quarters) is even. Centring is required when the moving average is calculated over an even number of periods, such as four quarters in a year. This is because the moving average calculated over an even number of periods falls between the time points of the original data. Centring the moving average aligns it with the original time points, which is necessary for calculating the ratio of the actual value to the moving average.

Correct Answer: C. The activity in that quarter is 42% above the average for the year. A seasonal index of 142 means that the activity in that specific quarter is 142% of the average activity per quarter for the entire year. Since the base is 100, an index of 142 signifies that the activity is 42% higher than the average (142 − 100 = 42).

Correct Answer: B. 7,100. The estimated number of boats rented out during the summer quarter is calculated by multiplying the average number of boats per quarter by the seasonal index for summer (expressed as a fraction of 100). The calculation is: 5,000 ∗ (142 ⁄ 100) = 5,000 ∗ 1.42 = 7,100 boats.

Correct Answer: A. To ensure that the seasonal indices sum up to the number of periods in a year, maintaining consistency for further calculations. For quarterly data, the sum of the seasonal indices should ideally be 400 (since there are four quarters and the average index is 100). If the sum of the modified means deviates from 400, an adjustment factor is used to scale the indices proportionally so that their sum equals 400. This ensures that the deseasonalised values, when multiplied back by the seasonal indices, accurately reflect the original scale of the data over a complete year.

Correct Answer: B. 0.9899. First, calculate the sum of the modified means: 91.25 + 107.70 + 113.25 + 91.90 = 404.10. The adjusting constant is calculated as (Desired Sum ⁄ Actual Sum). For quarterly data, the desired sum is 400. So, the adjusting constant is 400 ⁄ 404.10 ≈ 0.9899.

Correct Answer: D. Irregular Variation. Irregular variations are the unpredictable fluctuations in a time series that remain after accounting for trend, cyclical, and seasonal components. These are typically caused by sudden, random events that are not part of the usual pattern, such as natural disasters, strikes, or unexpected political events. For example, a sudden flood destroying crops would cause an irregular variation in agricultural output data.

Correct Answer: C. No, they are often difficult to isolate mathematically but their causes might be identifiable. Irregular variations are random and non-systematic, making them challenging to model and isolate using standard mathematical techniques applied to trend, seasonal, or cyclical components. While it’s difficult to predict when and with what magnitude they will occur, the specific events causing them (like a sudden change in government policy or a major accident) can often be identified after they happen.

Correct Answer: B. A severe and unexpectedly cold winter. Irregular variations are caused by unpredictable events. While population growth contributes to the long-term trend, and summer peaks are a seasonal variation, an unusually severe and unexpected cold winter would cause a sudden and irregular surge in heating-related electricity consumption that is outside the typical seasonal pattern.

Correct Answer: A. Autoregressive Integrated Moving Average (ARIMA) model. The text mentions Autoregressive Models (AR), Moving-average models (MA), and Autoregressive Moving Average (ARMA) models as basic models for time series forecasting. The ARIMA model, while a widely used time series model, is not listed as one of the three basic models in this specific section of the text.

Correct Answer: A. A linear combination of past values of the variable itself. An Autoregressive model is based on the idea that the current value of a time series can be predicted as a linear function of its past values. The term “autoregression” signifies that the regression is performed on the variable’s own prior observations.

Correct Answer: C. That it is a regression of the variable against itself (its past values). The term “autoregression” highlights the core concept of the AR model, where the value of a variable at a given time point is modelled based on a linear relationship with its values at previous time points. It’s a regression where the dependent variable is a past value of the series itself.

Correct Answer: B. A linear combination of error terms occurring contemporaneously and at various times in the past. The Moving-average model focuses on the error terms (the differences between the actual values and the model’s predictions). It posits that the current error is a linear combination of the current error and previous error terms in the series.

Correct Answer: C. Autoregressive Moving Average (ARMA) model. The ARMA model combines the features of both AR and MA models. It uses both past values of the time series (autoregressive component) and past error terms (moving-average component) to forecast future values.

Correct Answer: C. The impact of both previous lags and the residuals. The ARMA model integrates both the autoregressive (AR) and moving-average (MA) aspects. This means it considers both the linear relationship with past values of the series itself (lags) and the linear relationship with past error terms (residuals) to generate forecasts.

Correct Answer: B. 14.96. The trend value (y′) is calculated using the equation y′=18+0.16∗(−19)=18−3.04=14.96.

Correct Answer: A. 112.3%. The “Per cent of Trend” is calculated as (Deseasonalised Sales / Trend Value) * 100. Here, (16.8 / 14.96) * 100 ≈ 112.3%.

Correct Answer: B. 16.56. The trend value (y′) is calculated using the equation y′=18+0.16∗(−9)=18−1.44=16.56.

Correct Answer: B. 93.0%. The “Per cent of Trend” is calculated as (Deseasonalised Sales / Trend Value) * 100. Here, (15.4 / 16.56) * 100 ≈ 93.0%.

Correct Answer: B. 18.16. The trend value (y′) is calculated using the equation y′=18+0.16∗(1)=18+0.16=18.16.

Correct Answer: C. 116.7%. The “Per cent of Trend” is calculated as (Deseasonalised Sales / Trend Value) * 100. Here, (21.2 / 18.16) * 100 ≈ 116.7%.

Correct Answer: B. A set of observations collected at successive points in time or over successive periods. This is the standard definition of a time series, representing data points indexed in time order.

Correct Answer: B. To provide a basis for forecasting future values or patterns. Time series analysis uses historical data patterns to predict future occurrences.

Correct Answer: C. Secular Trend, Cyclical Fluctuation, Seasonal Variation, Irregular Variation. These are the four types of variations commonly identified and analysed within time series data.

Correct Answer: B. To simplify calculations involving time periods, especially in regression equations. Coding, such as subtracting the mean time, simplifies the numerical values used for time (like years), making computations easier.

Correct Answer: C. Patterns of change that repeat regularly within a one-year period. Seasonal variation refers to predictable patterns, like higher sales in a particular season, occurring within a year.

Correct Answer: D. Business cycles typically span periods longer than a year and are less regular than seasonal patterns. Business cycles involve fluctuations over multiple years (peaks and troughs), unlike seasonal variations which are annual and regular.

Correct Answer: C. Eliminating the identifiable, regular seasonal patterns from the series. Deseasonalisation is the process of removing the seasonal component to better analyse other components like trend and cyclical variations.

Correct Answer: C. To better isolate and analyse the trend and cyclical components of the series. By removing the seasonal effect, analysts can get a clearer picture of the underlying long-term trend and cyclical movements.

Correct Answer: B. To model the average long-term increase or decrease in the data over time. This equation represents the secular trend. For example, if a shop’s sales over 3 years are ₹100, ₹110, and ₹120 lakhs, a linear trend equation might approximate this steady growth, helping to understand the general direction even if some years had slight deviations.

Correct Answer: B. That the past trend and the factors influencing it will continue similarly into the future. Forecasting extends the historical pattern. For instance, if the equation shows sales increased by an average of ₹5 lakhs per year historically, forecasting for next year assumes a similar ₹5 lakh increase, barring major changes.

Correct Answer: B. The actual value expressed as a percentage of the expected trend value for that period. It shows how the actual data compares to the trend estimate. For example, if the trend equation predicts sales of ₹200 lakhs for a year (Trend Value), but actual sales were ₹220 lakhs (Actual Value), the Percent of Trend is (220 / 200) * 100 = 110%. This shows actual sales were 10% above the trend estimate for that year.

Correct Answer: C. The percentage deviation of the actual value from the estimated trend value, relative to the trend value. It quantifies the fluctuation around the trend. Using the previous example (Actual=₹220 lakhs, Trend=₹200 lakhs), the Relative Cyclical Residual is {(220 – 200) / 200} * 100 = +10%. This indicates the actual value was 10% above the trend, highlighting cyclical or irregular influences.

Correct Answer: B. When the rate of growth or decline in the data itself appears to be changing over time. A linear trend assumes constant change, while a parabola allows for acceleration or deceleration. For example, if new product sales grow slowly initially, then rapidly, and finally level off, a curve (parabola) would fit better than a straight line.

Correct Answer: C. 102.6%. The calculation is (Actual Value / Trend Value) * 100 = (195 / 190) * 100 ≈ 102.6%.

Correct Answer: A. +12.0%. The calculation is {(Actual Value – Trend Value) / Trend Value} * 100 = {(28 – 25) / 25} * 100 = (3 / 25) * 100 = 12.0%.

Correct Answer: C. 105.0%. Calculation: (Actual / Trend) * 100 = (21 / 20) * 100 = 1.05 * 100 = 105.0%.

Correct Answer: A. +9.7%. Calculation: {(Actual – Trend) / Trend} * 100 = {(32.9 – 30.0) / 30.0} * 100 = (2.9 / 30.0) * 100 ≈ 9.7%.

Correct Answer: C. The actual value is 5% lower than the trend estimate. A Percent of Trend less than 100 means the actual data point falls below the calculated trend line.

Correct Answer: B. The actual value is 8% lower than the trend value. A negative residual indicates the actual data point is below the trend estimate by that percentage of the trend value.

Correct Answer: B. 100.0%. For Year 3, x=2. Trend value = 50 + 1*(2) = 52. Actual value = 52. Percent of Trend = (52 / 52) * 100 = 100.0%.

Correct Answer: D. +7.8%. For Year 2, x=1. Trend value = 50 + 1*(1) = 51. Actual value = 55. Relative Cyclical Residual = {(Actual Value – Trend Value) / Trend Value} * 100 = {(55 – 51) / 51} * 100 = (4 / 51) * 100 ≈ 7.8%.}

Correct Answer: B. To measure the typical pattern of variation that occurs within a one-year period. Seasonal indices quantify the regular, predictable ups and downs related to seasons or specific times within the year (e.g., higher sales in Q4 due to festivals).

Correct Answer: D. Calculating a modified mean by discarding the highest and lowest index values before averaging the remaining ones. This technique, mentioned in the broader context of seasonal index calculation implied by Q13 and Q14a, helps to remove the influence of unusually high or low values caused by cyclical or irregular factors, providing a more representative seasonal measure.

Correct Answer: C. Removing the estimated seasonal influence from the original data. Deseasonalisation aims to strip away the predictable seasonal pattern to better view other components.

Correct Answer: B. Deseasonalised Value = Original Value / (Seasonal Index / 100). To remove the seasonal effect, the actual value is divided by the seasonal index (expressed as a proportion).

Correct Answer: C. To get a clearer view of the underlying trend and cyclical components. Removing the often-strong seasonal fluctuations makes it easier to visually identify the long-term direction (trend) and medium-term cycles.

Correct Answer: B. By dividing the deseasonalised data points by the corresponding trend values. This expresses the remaining variation (cyclical and irregular) as a ratio or percentage relative to the trend.

Correct Answer: B. Target Q1 = (₹84 lakh / 4) * (80 / 100). First, find the average quarterly sales (₹84 lakh / 4 = ₹21 lakh). Then, adjust this average using the seasonal index for Q1.

Correct Answer: C. ₹75 lakh. Average quarterly sales = ₹200 lakh / 4 = ₹50 lakh. Q3 Target = Average Quarterly Sales * (Q3 Index / 100) = ₹50 lakh * (150 / 100) = ₹50 lakh * 1.5 = ₹75 lakh.

Correct Answer: B. To ensure the average is aligned correctly in time with the original quarterly data points. A simple 4-quarter moving average falls between quarters. Centring (by averaging two adjacent moving averages) aligns the average with the actual quarter, enabling proper comparison.

Correct Answer: C. The underlying secular trend. Since the seasonal component has been removed, fitting a trend line to the deseasonalised data aims to capture the long-term direction (increase, decrease, or stability) of the series.