Econometrıcs II (ENG) — Ünite 3 Soru-Cevap

A time series is a set of observations in sequence by time in a specific time interval such as monthly, quarterly, and annually. On the other hand, time series analysis is the investigation of dependency at different time points. The purpose of time series analysis is to identify any recurring patterns in a time series, quantify these patterns through building a statistical model and then use the statistical model to prepare forecasts to estimate future values of the time series.

Time series analysis assumes that data values of a time series variable are determined by four underlying environmental forces that operate both individually and collectively over time. They are trend, cycle, seasonality, and random (irregular) variation. Time series analysis attempts to decompose each of these components and quantify them statistically. This process is known as decomposition of the time series. Then, these components are combined and used to forecast the time series variable.

Exponential smoothing is a time series method for forecasting univariate time series data. Exponential smoothing is generally used to make forecasts of time-series data based on prior assumptions by the user, such as stationarity, seasonality or deterministic trends.

Trend: It indicates the long-term tendency of either upward or downward movement in the average value of the forecast variable Yt over time. It can be assumed that variables have a trend when they show a constant increase or decrease over time, such as population and income, or do not show a sudden change against any effect.

Cyclical Movements: The oscillations that occur with upward and downward around the long-term trend curve indicate the effect of cyclical movements. Business cycles caused by expansion or contraction in an economy over time create cyclical movements. There are several reasons why cyclical movements occur in a time series. For instance, structural changes in the economy (monetary or fiscal policies) or changes in the demand for any product over time such as a decrease in the sales of large cars as a result of the increase in fuel prices may cause cyclical movements

Seasonal Change: Permanent fluctuations in a specific time period within a year, such as monthly or quarterly, can indicate seasonal changes in time series. For instance, ice cream and soft drink sales are much higher in summer than in winter months, sales in stationery stores can be much higher in September the opening month of schools, the amount of sales of agricultural products differing between summer and winter periods can indicate the presence of seasonal variation. Seasonal change is also a cyclical movement, but it has predictable features because it follows a much more regular course.

Random Variation: It is the component that reflects the upward and downward oscillations in a time series after decomposing the effects of trend, cyclical movements and seasonal variation. The random component refers to the observed but unpredictable changes in time series as a result of random events that occur because of unforeseen (unexpected) movements such as political events, natural disasters, war or famines.

Trend analysis refers to the examination of long-term changes in the level of a time series. It focuses on identifying and understanding the underlying patterns that exhibit a consistent direction over an extended period. Time series trend analysis helps reveal whether the data is experiencing a gradual increase (upward trend) or decrease (downward trend) in its overall level.

If the time series Yt shows a constant absolute increasing/decreasing trend in every time period, it can be considered that the linear trend model is suitable. The linear trend model is written as:

Yt = α + βt

Here;

Yt: The observation value at time t

t: Time variable

α ve β Parameters of the model

The time variable is a code variable that assign the value 1 for the first period and then codes are assigned all successive values consecutively increasing integer numbers in the following periods (t = 1, 2, ..., n). The starting point for the time variable is not necessarily to be a value of 1. You can start with a negative number or a positive number greater than 1, but the codes must be integers that increase one by one.

If the time series Yt changes with constant percentage increases or decreases instead of constant absolute increases or decreases, the appropriate model is the exponential trend model. The exponential trend model is described as follows:

Yt = αeβt

This model can be expressed in the linear form by taking its natural (base-e) logarithm transformation

ln(Yt ) = ln(α) + βt

If defined as ln(Yt ) = Yt * and ln(α*), the above equation can be written in linear form as:

Yt* =α * +β t

This model is also called Log-Linear or semi-logarithmic model. In the exponential trend model, the coefficient β is interpreted as the average growth rate. At the same time, it can be described as follows:

ln(Yt ) - ln(Yt -1) = ln(α) + βt - [ln(α) + β(t-1)] = β

If the error term that disturbs the deterministic relationship is included into the linear trend model, the equation will be as follows:

Yt = α + βt + εt

In this equation, εt represents the error term. The assumptions for the error term are as follows:

1. The error term is a random variable.
2. The mean (expected value) of the error term is zero (E (εt) = 0).
3. The variance of the error term is constant (Var (εt ) = σε2 ).
4. There is no serial correlation (autocorrelation) between the error terms. Cov (εt , εt - j ) = 0 for j = 1, 2, ...
5. There is no relationship between the error term and the trend variable (Cov (εt , t) = 0).
6. The error term has a normal distribution with zero mean and constant variance. Thus, logarithmic first-order differences correspond to the growth rates for the time series.

Least squares is a method for parameter estimation based on the minimization of the sum of squared errors. Given the time variable (t), the conditional expected value of Yt , which is in the position of the dependent variable, becomes E(Yt ⎜t) = α + βt according to assumption 2, E (εt ) = 0. This conditional expected value can be considered as the estimated value for the Yt time series. ˆYt as the estimated value of Yt would be ˆYt = ˆα + ˆβt . The error term can be defined as the difference between the actual (Yt ) and the predicted ( ˆYt ) values.

The most commonly used model selection criterion in practice is the root mean square error (RMSE). In order to use this criterion, in addition to having the same number of variables on the right side of the related equation, errors need to be measured in the same way by (Yt - ˆYt ).

The calculation method for the RMSE criterion is as follows:

RMSE= Σ (Yt –ˆYt )2 n

The most widely used unit free measure is the mean absolute percentage error (MAPE),

MAPE=100 1n |Yt –ˆYt | |Yt | Σ

If the units of error values vary, for example, if one forecast model uses the actual values and another forecast model uses the values with the natural logarithm, the statistics that can be used are the “Mean Absolute Percentage Error” (MAPE). The MAPE statistic eliminates the disadvantages that can arise when comparing models with different unit values. However, a disadvantage of the MAPE is that it is relevant only for ratio-scaled data. The model with the forecast evaluation criteria value closest to zero is the most suitable. In other words, it would be appropriate to choose the model with the minimum forecast evaluation criteria value.

A moving average removes the short-term fluctuations in a time series by taking successive averages of groups of observations. Some of the time series that are measured with a frequency throughout the year, such as monthly or quarterly, include seasonal variation components. Seasonality is a common characteristic of time series. Seasonality can appear in two forms: additive and multiplicative. In the former case, the amplitude of the seasonal variation is independent of the level, whereas in the latter it is connected. 

Moving averages are used in the calculation of seasonal indices as well as in producing a time series that does not contain seasonal variation and has very little random variation (irregular movement) because the moving average series are much smoother and allow the trend or cyclicality of the series to be seen quite clearly.

Another approach taken in the process of smoothing and predicting a time series is exponential smoothing methods. Unlike moving averages, these methods use all time series observations to determine smoothed values for any time period. In this section, exponential smoothing methods will be discussed.

For the simple exponential smoothing method to be suitable, the relevant time series data must be stationary. Beside a time series mean and variance remain constant with respect to time, the autocovariance between two different time points such as and depends not on the real times and , but on the time difference ⎜t - k⎜ = s. Thus, it is called a weak stationary time series. If the relevant time series is weakly stationary, forecasts made with the simple exponential smoothing method will better capture the original time series.

Seasonal variations are fluctuations in a time series that are repeated at regular intervals within a year (e.g., monthly, quarterly). Seasonal variations are caused by recurring environmental influences, such as climatic conditions (the seasons), and special recurring events (e.g., annual religious holidays, public and school holidays). For time series with seasonality, it would be appropriate to make forecasts with Holt- Winter’s linear and seasonal exponential smoothing method. The so-called three-parameter method Holt- Winter’s process takes into account not only past observations and trend forecasting, but also smoothed seasonal factors for each time period.

In the Holt-Winter method, initial seasonal factors, initial trend prediction, and initial smoothing value for the last period of the previous year (the last quarter for quarterly data, and December for monthly data) are needed from the observation interval.

There are two processes followed in finding these values. In the first process;

1. The initial seasonal factors are taken as 1. If the data is quarterly, the initial seasonal factor values will be F0 = F-1 = F-2 = F-3 = 1.
2. The initial trend estimate b0 is taken equal to zero.

iii. The initial smoothing value (S0) is selected equal to the actual value for the last period of the first year. This value is the last quarter of the year preceding the observed time frame for quarterly data. For instance, if the data range consists of quarterly observations between 2016 and 2023, the value (S0) corresponding to the last quarter of 2015 will be equal to the actual value for the fourth quarter of 2018. Then, it will be as S0 = Y4.

In the second process, the following steps are followed:

1. Seasonal factors are calculated using data from the first two years (or more). These values will be seasonal factors for the periods of the year preceding the data range.
2. The data for the first two (or more) years are seasonally adjusted by the seasonal factors calculated in step-1. Let’s define the seasonally adjusted time series as dt. By using these data, dt = α + βt + εt linear trend model parameters are estimated by least squares method, and b0 = ˆβ value is taken into account as the initial value for the trend.

iii. The initial smoothing value (S0) is ˆα the intercept coefficient of the linear trend model whose parameters are estimated with the seasonally adjusted data in Step-2, and the seasonal factors calculated in Step-1, corresponding to the most recent period (the last quarter for quarterly data, and December for monthly data) is found by multiplying the seasonal factor value.

There are two major definitions of index numbers. Wheldon defines it as follows “Index number is a statistical device for indicating the relative movement (dynamics) of data where measurement of actual movement is difficult or incapable of being made”; and according to Edgeworth “Index number shows by its variations (being dynamical) the changes in a magnitude which is not susceptible either of accurate measurement in itself or direct valuation in practice” (Samuel, Ovie, 2015).

Index numbers may be broadly classified into various categories such as price, quantity, and value. Only price indexes will be discussed in this section. Price index numbers are a statistical measurement based on the relative changes in the prices of a particular commodity like silver, gold, iron, etc. or a group of commodities like consumer goods, cereals, dairy products, etc. Index numbers are proportional values that express percentage changes and are unit-independent.

If a single commodity is used to calculate the index number, it is called a simple price index, and if two or more commodities are used, it is called a composite (or aggregative) price index. To measure the changes in price over time, two different time periods can be defined in calculating price index number. These are fixed base period and chain base period. If the index numbers are calculated according to a base period, it is called fixed, and if it is calculated according to the previous period, it is called as a chained index. The choice of the fixed period is important in calculating the index numbers with the fixed base. If possible, the base period should be determined as the period of economic stability. In other words, periods of economic overgrowth or recession or economic crisis periods should not be chosen as the base period.

An index number measures the percentage change from a base period, which has an index value of 100. Index values above 100 indicate an increase in the level of activity being monitored, while index values below 100 reflect a decrease in activity relative to the base period.

If the index value is greater than 100, 100 is subtracted from the index value and this value is interpreted as a percentage increase. On the other hand, if the index value is less than 100 the index value is subtracted from 100 and this value is interpreted as a percentage decrease. In the calculation of the fixed-base index, the interpretation of the index number for the current period is made according to the fixed period, while in the chain index it is made according to the previous period.