Time Series Analysis and Forecasting

Coding Time Variable by Taking Origin at the Beginning

Suppose we have time-series data for the years 1990, 1991, 1992, and 1994.

We can take the origin of a time series at the beginning and assign $x = 0$ to the first period and $1, 2, 3, …$ to other periods. The code for the year will be

Coding Time Variable by Taking Middle Years as Zero

To simplify the trend calculations, the time variable $t$ (year variable) is coded by taking deviations $t-\overline{t}$, where $\overline{t}$ is the average number computed as $\overline{t}=\frac{First\, Period + Last\, Period}{2}$. Taking $x=t-\overline{t}$ we get
$$\sum x = 0 = \sum x^3 = \sum x^5 = \cdots$$

There are two cases when coding a Time Variable (when taking zero in the Middle):

• When there are an odd number of Years:
  For an odd number of years (as in the period 1990 to 1994) the $\overline{t}$ is the middle point. The $\overline{t}$ is $\overline{t} = (1990+1994)/2=1992$ the code for the year $t$ is $x=t-\overline{t}$. For t=1990, we have $x=1990-1992 =0$. Thus the coded year is zero at $\overline{t}$. Now after taking x=0 at the middle of an odd number of years, we assign $-1, -2, …$ for the years before the middle of the year and $1,2,…$ for the years after the middle year.

  Year (t)	$x=t-\overline{t}$
  1990	-2
  1991	-1
  1992	0
  1993	1
  1994	2

• When there are even numbers of years
  Suppose we have time-series data for the years 1990, 1991, 1992, 1993, 1994, and 1995. The value of middle point is $\overline{t} = (1990+1995)/2 = 1992.5$. So $x=0$ halfway between the years 1992 and 1993 (in the middle of 1992 and 1993). For $t=1992$, we have $x=t-\overline{t}=1992-1992.5=-0.5$. Thus coding the middle of an even number of years as $x=0$, we assign $-0.5, -1.5, -2.5, …$ for the years before the middle year and $0.5, 1.5, 2.5, …$ for the years after the middle year as shown below

Year(t)	$x=t-\overline{t}$	$x=\frac{t-\overline{t}}{1/2}$
1990	-2.5	-5
1991	-1.5	-3
1992	-0.5	1
1993	0.5	1
1994	1.5	3
1995	2.5	5

To avoid decimals in the coded year, we can take the unit of measurement as $\frac{1}{2}$ year. Therefore, after coding $x=0$ in the middle of an even number of years, we assign $-1,-3, -5,…$ for the year before the middle year and $1,3,5,…$ for the years after the middle year as shown above.

Read more about Coding Time Variables in R

R Programming Language

Computer MCQs

Here we will discuss the multiplicative models and Additive Models.

The analysis of a time series is the decomposition of a time series into its different components for their separate study. The process of analyzing a time series is to isolate and measure its various components. We try to answer the following questions when we analyze a time series.

1. What would have been the value of the variable at different points in time if it were influenced only by long-time movements?
2. What changes occur in the value of the variable due to seasonal variations?
3. To what extent and in what direction has the variable been affected by cyclical fluctuations?
4. What has been the effect of irregular variations?

The study of a time series is mainly required for estimation and forecasting. An ideal forecast should be based on forecasts of the various types of fluctuations. Separate forecasts should be made of the trend, seasonal, and cyclical variations. These forecasts become doubtful for a forecast of irregular movements. Therefore, it is necessary to separate and measure various types of fluctuations present in a time series.

A value of a time series variable is considered as the result of the combined impact of its components. The components of a time series follow either the multiplicative or the additive model.

Fro both Multiplicative and additive models, let $Y$= original observation, $T$= trend component, $S$=seasonal component, $C$=cyclical component, and $I$=irregular component.

Multiplicative Models

It is assumed that the value $Y$ of a composite series is the product of the four components. That is

$$Y = T \times S \times C \times I,$$

where $T$ is given in original units of $Y$, but $S$, $C$, and $I$ are expressed as percentage unit-less index numbers.

Additive Models

It is assumed that the value of $Y$ of a composite series is the sum of the four components. That is

$$Y = T + S + C + I,$$

where $T$, $S$, $C$, and $I$ all are given in the original units of $Y$.

Time series analysis is the analysis of a series of data points over time, allowing one to answer a question such as what is the causal effect on a variable $Y$ of a change in variable $X$ over time? An important difference between time series and cross-section data is that the ordering of cases does matter in time series.

Rather than dealing with individuals as units, the unit of interest is time: the value of $Y$ at time $t$ is $Y_t$. The unit of time can be anything from days to election years. The value of $Y_t$ in the previous period is called the first lag value: $Y_{t-1}$. The jth lag is denoted: $Y_{t-j}$. Similarly, $Y_{t+1}$ is the value of $Y_t$ in the next period. So a simple bivariate regression equation for time series data looks like: \[Y_t = \beta_0 + \beta X_t + u_t\]

$Y_t$ is treated as a random variable. If $Y_t$ is generated by some model (Regression model for time series i.e. $Y_t=x_t\beta +\varepsilon_t$, $E(\varepsilon_t|x_t)=0$, then ordinary least square (OLS) provides a consistent estimates of $\beta$.

See the YouTube video about Multiplicative and Additive Models.

Selection between Multiplicative and Additive Models

A question arose about how to Choose Between Multiplicative and Additive Models. The additive model is useful when the seasonal variation is relatively constant over time. When the seasonal variation increases over time, the multiplicative model is useful.

Read about Introduction to Time Series Data

Learn more about Multiplicative and Additive Models

Frequently Asked Questions about R Programming Language and R Data Analysis

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