97joseph/GDP_R_Modelling

From mathematics to coding models

GDP MODEL FOR A MATHEMATICAL MODEL

From mathematics to coding models

This program finds the expected value of life satisfaction based on the inputted value of country's GDP. The Life satisfaction is within the range 0 to 10, 10 being the highest. Make sure to change the address of filename depending upon where you store it in your computer.

Structure of the Model

model		a list specifying an ARIMA model. The componentsar and ma, if present, should be numeric vectors giving the autoregression and moving average parameters for the model. If you also want a difference parameter, supply the order component, a three-long integer vector giving the length of the ar component, the number of differences, and the length of the ma component, respecively. (If supplied, the first and last elements of order must match the lengths of the ar and ma components.)
n		the length of the series to be simulated.
innov		a univariate time series or vector of innovations to produce the series. If not provided,innov is generated using rand.gen. Missing values (NAs) are not allowed. If provided, its length must match the n argument.
n.start		the number of start-up innovations. The start-up innovations are generated byrand.gen if start.innov is not provided. n.start must be as long as ar + ma.
start.innov		a univariate time series or vector of innovations to be used as start up values. Missing values (NAs) are not allowed. These are transformed by the Choleski "square root" of the correlation matrix corresponding to the model so the simulated process begins with something close to its stationary state if you supply iid (independent and identically distributed) innovations.
rand.gen		a function that is called to generate the innovations. Usually,rand.gen is a random number generator.

BACKGROUND

The Gross Domestic Product (GDP) is that the value of all product and services made at intervals the borders of a nation in an exceedingly year. In this paper, the Box-Jenkins approach has been used to build the appropriate Autoregressive-Integrated Moving-Average (ARIMA) model for the Egyptian GDP data. Egypt’s annual GDP data obtained from the World-Bank for the years 1965 to 2016. We find that the appropriate statistical model for Egyptian GDP is ARIMA (1, 2, 1). Finally, we used the fitted ARIMA model to forecast the GDP of Egypt for the next ten years. Keywords: Box-Jenkins approach, Egypt, Forecasting, Goodness-of-fit measures, Gross domestic product, Residuals analysis.

DEFINITION

GDP represents the market value of all goods and services produced by the economy during the period measured, including personal consumption, government purchases, private inventories, paid-in construction costs and the foreign trade balance (exports minus imports). GDP can be measured in three ways: (i) the expenditure approach, (ii) the production approach, and (iii) the income approach. The issue of GDP has becomes the most concerned amongst macro economy variables and data on GDP is regarded as the important index for assessing the national economic development and for judging the operating status of macro economy as a whole (Ning et al, 2010). It is often considered the best measure of how well the economy is performing. Also, it is a vital basis for government to set up economic developmental strategies and policies.

MATHEMATICAL FRAMEWORK

The time series analysis can provide short-run forecast for sufficiently large amount of data on the concerned variables very precisely, see Granger and Newbold (1986). In univariate time series analysis, the ARIMA models are flexible and widely used. The ARIMA model is the combination of three processes: (i) Autoregressive (AR) process, (ii) Differencing process, and (iii) Moving-Average (MA) process. These processes are known in statistical literature as main univariate time series models, and are commonly used in many applications. 2.1. Autoregressive (AR) model An autoregressive model of order p, AR (p), can be expressed as: 𝑋𝑡 = 𝑐 + 𝛼1𝑋𝑡−1 + 𝛼2𝑋𝑡−2 + ⋯ + 𝛼𝑝𝑋𝑡−𝑝 + 𝜀𝑡 ; 𝑡 = 1,2, … 𝑇, (1) where 𝜀𝑡 is the error term in the equation; where 𝜀𝑡 a white noise process, a sequence of independently and identically distributed (iid) random variables with 𝐸(𝜀𝑡 ) = 0 and 𝑣𝑎𝑟(𝜀𝑡 ) = 𝜎 2 ; i.e. 𝜀𝑡 ~𝑖𝑖𝑑 𝑁(0, 𝜎 2 ). In this model, all previous values can have additive effects on this level 𝑋𝑡 and so on; so it's a long-term memory model. 2.2. Moving-average (MA) model A time series {𝑋𝑡 } is said to be a moving-average process of order q, MA (q), if: 𝑋𝑡 = 𝜀𝑡 − 𝜃1𝜀𝑡−1 − 𝜃2𝜀𝑡−2 − ⋯ − 𝜃𝑞𝜀𝑡−𝑞.