A Discrete Time Markov Chain Model for the Assessment of Inflation Rate in Pakistan

Markov chains epitomize a class of stochastic process for a wide range of applications. Specifically, discrete time Markov chains (DTMC) is employed to model the transition probabilities between discrete states with the help of the matrices. To examine and forecast the time series the Markov chain model is applied. The most important indicator in macroeconomics is inflation, which persisted in double digits in 1970s and also in last several years. Different states are checked with the model by using inflation rate data form July 2000 to April 2015. A simulation technique used for random sequences of inflation predefine states for one year and take 1 st quarter data from it and then model the estimates by maximum likelihood and maximum likelihood with Laplace smoothing methods and check the equilibrium distribution using both techniques. Estimates obtained by Laplace Smoothing technique are reliable because it control the variation on the Maximum Likelihood estimates.


I Introduction
Discrete Markov chain models are valuable for demonstrating mostly applied structures such as trade systems, line up systems [II] and account systems [III]. These models are used for displaying definite data series can also be found in mostly real world areas [IV]. Time series are used often in mostly real world areas. If anyone modeled the time series precisely, then it is easy to make accurate estimates and also optimally forecast in a decision procedure [V]. A discrete time Markov chain is a random procedure that undertakes transitions from one state space to another state space. It must hold a "memory lessly", in which the chances of the next state depend on the current state not on the system of events that goes before. This case of the Memory lessness is called Markov property (MP).In real world processes, statistical models MC have many applications. A MC is a stochastic procedure with the MP. The term "MC" refers to the arrangement of arbitrary variables such a procedure exchanges through, with the MP significant sequential dependence only between neighboring periods. It can thus be used for describing systems that follow a chain of linked events, where what happens next depends only on the current state of the system. Different types of MP are selected as "MC" in the previous literature. Generally the stretch is kept for a process with a discrete set of times is called discrete time Markov chain [VI]. MC refers to a continuous deprived of explicit indication. Although the time is taken as a discrete parameter, there is no restriction on the state space in the MC, the period may mention to a method on a random state space [VIII]. Although in MC applications finite or infinite states are considered as discrete state spaces that have more statistical analysis, moreover time catalogue and state space restrictions, a lot of disparities additions and simplifications are founded. In economics, inflation is a continued fluctuation in the general price level of goods and services in an economy with respect to a period of time. Inflation delivers important vision on the national economy and occurs in any economy but with a different rate and strength. Inflation in the price of Food is a highest issue being handled by emerging states a like Pakistan. This indicator use excessive quantity of compression with the economic circumstances of any state [VII]. The movable economic and financial policies of the government of Pakistan have caused in upgrading in numerous macroeconomic factors with Gross Domestic Product (GDP) growth in several years. For example in 1960s, 1980s and the few centuries of the first era of the 21 st century, this persisted above 6 percent during 2004-06. Despite this impressive performance of the economy, some worrisome factors have also seemed on this section. The most substantial of these indicators is inflation, which persisted in dual number in 70s and also in last numerous years [I].

II Methodology
A growth in import prices was also cause a salient indicator in producing inflation. Correspondingly, some researchers suspected that indirect taxes are the main reasons of inflation. The Wheat Support Price (WSP) recognized as an important element of inflation in Pakistan (Khan and Qasim, 1996). For checking the variability in the inflation rates, the inflation rate data is taken on monthly basis form July-2000 to April-2015 from state bank of Pakistan. Then differentiate inflation rate into 3 states, if rate<=0 then it categorize as "Deflation", if <=1 then The set of possible states , , , … , of Xn can be finite or countable and it is named the state space of the chain (Ching.Wet al., 2008).Firstly build a Markov chain model for a pragmatic definite data sequence, adopt the following canonical form representation: 0,1,0 1,0,0 0,0,1 . .. 0,1,0 For X0=2, X1=1, X2=3 … Xn=2.
To estimate the transition probability matrix for the above perceived Markov chain, considered the following simple measures. By totaling the transition frequency from State ito State jin the arrangement, build the transition frequency matrix N then the transition probability matrix P for the arrangement as follows:

⎣ ⎦
After making the transition matrix check the distribution of the states by using different initial states and check the stability situation after n generation of distribution of the states. In other words to find out the stationary distribution and identifying absorbing and transient states. For statistical analysis simulating a random sequence from an underlying DTMC. Then checks the estimates by using two methods, maximum likelihood, and maximum likelihood with Laplace smoothing.
The maximum likelihood estimator (MLE) of the entry, where the nij element consists in the number sequences (Xt = Si; Xt+1 = Sj) found in the sample, that is The Laplace smoothing approach is a variation of the MLE. For tackle the variation in MLE, Laplace smoothing approach will be used. Where the is substituted by ( ∝. Here ∝an arbitrary positive stabilizing parameter. 2058824 This matrix show that the probability of inflation rate from creeping to creeping is 48.23%, 20% creeping to galloping and 31.76% of creeping to trotting. Similarly the probabilities of inflation rate going from galloping to creeping is 55.88%, 23.52% from galloping to trotting, and 20.58% from galloping to galloping. The probability inflation rate of going from trotting to creeping is 41.37%, from trotting to trotting is 41.37%, and 17.24% from trotting to galloping inflation.

Figure 1: I-graph of Transition Probability Matrix.
For the graphical presentation of transition matrix in Markov chain model igraph is used, now according to the graph we can easily explain the probability of inflation rate moves from one state to another state. Let Creeping = C, Galloping = G, Trotting = T. this graph shows that the occurrence of inflation rate from C to C is 48%, 20% creeping to galloping and 32% creeping to trotting. The probability of G to C is 56% G to G is 21% and 24%  Vol.9, No.5, 2019 chances of inflation rate from G to T inflation. Similarly the chances of inflation rate from T to C is 41% , 17% from T to G and 41% chance of inflation rate moves from T to T.

III.I. Equilibrium Distribution of Transition Matrix
First chose [1, 0, 0] this initial state in which 1 st state of inflation rate is present others are absent then calculate its initial probability vector which is [0.4823529, 0.2000 0.3176471]. This is also called the generation of probability vector which explains that there are 48.23% probability of creeping state and 20% probability of galloping and 31.76% probability of trotting state. The entire probability vectors are generated by this formula.

III.II. Equilibrium Distribution of Transition Matrix
First chose [1, 0, 0] this initial state in which 1 st state of inflation rate is present others are absent then calculate its initial probability vector which is [0.4823529, 0.2000 0.3176471]. This is also called the generation of probability vector which explains that there are 48.23% probability of Creeping state and 20% probability of Galloping and 31.76% probability of Trotting state. The entire probability vectors are generated by * this formula. 0.4741333 0.1919165 0.3339502 By using this initial probability vector generates next probability vectors and checks the stability in which generation inflation states probability vectors are stable. After 7 th generation the probability vector of inflation states shows the stable chances. We can be see it from the above table the probability vectors of 7 th , 8 th and 9 th generation are stable at specific probability vector, which explain that there are 47.41% chance of Creeping, 19.19% of Galloping and 33.39% of Trotting state. All the next generations will show the same chances of occurrences of the inflation states, we may generate it into 10 th , 11 th times or so on many times.

III.III. Conditional Distributions
Conditional distribution is used for checking the probability of different states of inflation rate if any one of state is given. In the conditional distribution of inflation states, given that current inflation state is Trotting. The conditional distribution of inflation states, given that current inflation state is Galloping. The conditional distribution of inflation states, given that current inflation state is Creeping.

III.IV. Steady state
If the Markov chain is a time homogeneous Markov chain, so that the process is described by a single, time independent matrix,p ./ then the vector0 is called a stationary distribution. Note that there is no assumption on the starting distribution; the chain converges to the stationary distribution regardless of where it begins. Such 0 is  0.4741333 0.1919165 0.3339502 By using this initial probability vector generates next probability vectors and checks the stability in which generation inflation states probability vectors are stable. After 7 th generation the probability vector of inflation states shows the stable chances. We can be see it from the Table-4.12 the probability vectors of 8 th , 9 th and 10 th generation are stable at specific probability vector, which explain that there are 47.41% chance of Creeping, 19.19% of Galloping and 33.39% of Trotting state. All the next generations will show the same chances of occurrences of the inflation states, we may generate it into 11 th , 12 th times or so on many times.  Vol.9, No.5, 2019 initial probability vector which is [0.4736565 0.3421035 0.18424]. This is also called the generation of probability vector which explains that there are 18.42% probability of Trotting state and 47.36% probability of Creeping and 34.21% probability of Galloping state. The entire probability vectors are generated by * this formula. 2116496 By using this initial probability vector generates next probability vectors and checks the stability in which generation inflation states probability vectors are stable. After 7 th generation the probability vector of inflation states shows the stable chances. We can be see it from the above table the probability vectors of 8 th , 9 th and 10 th generation are stable at specific probability vector, which explain that there are 21.16% chance of Trotting, 41.42% of Creeping and 37.40% of Galloping state. All the next generations will show the same chances of occurrences of the inflation states, we may generate it into 11 th , 12 th times or so on many time.

VII: Conclusion
The current study based on Markov Chain Model, core purpose of present study was according to Markov Chain Model related to the economics features. For this study Macroeconomic indicator inflation rate is used and found that it is changing since several years. It has a foremost outcome on the progress and economic development of any country. Study data was form July 2000 to April 2015. we categorized our data into three different states that are "Trotting", "Creeping" and "Galloping" inflation rate as per as the key features of Markov Chain Model. First of all Checked the Transition count and Transition Matrix where it is easily to checked the progress chances of one state in comparison to another state. Equilibrium distribution is very important for the Markovian Transition Matrix its clarifies the stability point for the model whether it is checked on different initial vectors. Conditional distribution explained about the chances of other states if anyone is fixed that we know about it already then check the chances of there. The "markov chain" I graph R software package that helped for the Graphical presentation of the model. Transition Matrix are Smoothing Techniques that provided by using Maximum Likelihood and Maximum Likelihood with Laplace. Equilibrium distribution is obtain of the model separately by mutually techniques and checks the stability point of these distributions by taking different initial vectors. At the last the comparison is take place in stability points between two techniques. Estimates obtain on the fact and figures of Laplace Smoothing technique are reliable because it control the variation on the Maximum Likelihood estimates. It takes all the uncounts that are not counted in using Maximum Likelihood estimation.