Now in this video we will be discussing about stationarity and non stationarity. The random walk hypothesis the different techniques that are adopted to convert a non stationary data to stationary data first let's discuss about the concept of stationary stationary time series more or less structured around a given value which for longer series is often close to a historical average structure of changes may cause shifts in this value random processes with constant mean and variance over time are also stationary. A stationary series can be said to be a flat looking series without trend and it has a constant variance over time a constant autocorrelation over time and no periodic fluctuations intuitively stationarity of the series means that a certain type of statistical equilibrium is achieved as the distribution of the series does not change much it makes sense in practice because it provides a framework in which averaging makes sense.

So a stationary time series is one whose statistical properties such as mean variance, autocorrelation, etc are all constant over time, our time series is called stationary. If it doesn't wander off to infinity or it stays around the mean a stationary time series does not have any trend here the stationary time series is graphical present it it looks like this, it does not have any trend and the mean and variance of such time series is always constant. Now let's talk about the concept of non stationary non stationary series usually it has a strength therefore, they also have a trend in their mean and variance. A series with frequent level ships is also non stationary. So long sessions series is absolutely opposite of what is stationary and most time series are either increasing or decreasing or volatile or cyclical. Their mean and variance are all time dependent that is they are not constant.

For example, if we'll talk about increasing trend or decreasing trend, trend we can talk about GDP level and unemployment level. If we talk about volatile trend or cyclical trend, we can talk about stock market rises the trend for stock market prices the trend for oil prices. So, therefore, non stationary time series institution time series the statistical measures such as mean standard deviation autocorrelation shows the decreasing or increasing trend over time such time series data has a trend. So, here I have represented the non sufficient time series graphically. To see here the mean and variance both are not constant. Now let's move to the concept of random walk hypothesis.

Now, what is random walk hypothesis the random walk theory or the random walk hypothesis is a mathematical model of the stock market. That is the concept of random walk hypothesis has evolved around the stock market proponents of the theory believe that the prices of securities in the stock market evolve according to a random walk. Random Walk is a statistical phenomena where a variable follows no discernible train, trend and move seemingly After random, the random walk theory as applied to trading most clearly laid out by Burton Malkin, an economics professor at Princeton University posits that the price of securities moves randomly. And that therefore any attempts to predict future price movement either through fundamental or technical analysis is few times now there are two basic assumptions of the random walk theory, the random walk theory assumes that the price of each security in the stock market follows a random walk.

The random walk theory also assumes that the movement in the price of one securities independent of the movement in the price of another security So, random walk hypothesis is a type of non stationary process here the value of the time variable in the TF period is dependent on the value of the time variable in the T minus one period as well as on their error term. And the value of the time variable in the T minus one period is dependent on the time variable of t minus two periods and error terms of t minus one is period Which implies that the TF period time variable is dependent on the error term of the T minus one time period does over time the error terms increases gradually. So, the predictive power gets reduced, that is the predictive power of the model reduces and therefore, there is maximum or unexplained variation of the dependent variable.

Now, there are two types of random walk one is random walk without drift that is there is no intercept term or constant term in the random lock equation. Another is random walk with drift that is there is a drift parameter here delta is the drift parameter or the intercept the equations of random walk on is follows x equals to roll Stephens one plus M of t where if the mod rule was absolute value of rho is less than one then the value of XT will die down to zero which implies stationary when rho is equal to one we get a non stationary process which is the equation of random walk it is as follows x is equal to XT minus one plus error term of T that is when rho value becomes equal to one then this is the random walk equation. Now, attached is Done to check the stationarity of the data.

This test is called a df test or augmented Dickey fuller test the hypothesis for this test are as follows my null hypothesis is the data is non stationary and my alternative hypothesis is the data is stationary. Now, let's talk about the techniques that we need to adopt to convert non stationary data to stationary in time series It is very important to convert a non structured data to stationary because until unless we do not convert data into stationarity, we cannot do the forecasting for forecasting we need to convert the non stationary data to stationary So, these are the techniques that we can apply first technique is d trending we use this technique to convert the non stationary time series data to stationary time series data suppose our time series equation is YT equal to a plus bt present here you can denote as alpha also where bt is a time variable at p time point and up or mu p is the error term known as white noise white noise error terms follows normal distribution with constant mean and variance that is mean zero leadings one now in order to return the given model, we have to subtract the trend component that is alpha plus beta t from the dependent variable right so we get there at of UT which follows normal distribution with zero mean and constant sigma square is variance which is one that is heavy it is one which is constant does Titian entities attend does the detainee equation will be YT minus alpha plus beta t this is my trend component and I'm left with new t which is following normal distribution with zero and one that is zero mean and one is the variance and these are called ripeness error terms.

And in this way, the time series data is now stationary. differencing is another technique of converting non stationary data to stationary the concept of differentiation is as follows. Let's understand the concept of differencing using an equation so YT equals two YT minus one plus nu t wherever it is the present value and YT minus one is the past that is YT is the value of the dependent variable at the time period and YT minus one is the value of the dependent variable as the T minus a minus time period now YT minus right to men is equal to mx up which is a stationary process if we write del right To muted in this is first order differencing del vitamins we are differentiating this vitamin is YT one YT minus one is we are doing the first order differencing of it that is equal to mu t means, we are left with the error terms, which are whiteness error terms, which is constant mean and variance, because we have achieved stationary awkward in the first run of first order differences, but, suppose, if the data is still non stationary in nature that is the answer she is still not converted to stationary, after doing the first order differential we can go for the second order differencing where we the equation is represented as del square it is called vitamins vitamins can minus YT minus one minus YT minus two equal to empty their square varieties were two YT minus YT minus one can be denoted as del YT minus YT minus one minus YT minus two can be denoted as del YT minus one which is equal to UT.

So, after do the second order differencing we have got represent items with constant mean and variance that is it is following the normal distribution with constant mean and variance. stationarity can also be achieved if you take a look transformation now in this video we will be learning till here for now let's end this video over here. Goodbye. Thank you see you all for the next video