Lecture 22

[Music] in this video we discuss Arma process which is simply a combination of AR and Ma processes we also discuss how to identify the Arma process with the help of ACF and pacf diagrams next we discuss in a step-by-step manner how to build an Arma model for FOC casting we also discuss the prediction process with time series models and compare it with the prediction process with structural models with a simple example we discuss how to forecast with herma models we also discuss forecasting terminologies with one step ahead and multi-step ahead forast and rolling versus recursive sample forecasting we also discussed how to determine the accuracy of out of sample forecast in this video we'll introduce ARA which is autor regressive AR and moving average which is me so in this video we'll introduce ARA process which is a very important component of ARA models ARA class of models just to make a distinction here I here is nothing but the integration of the process so depending upon the levels of non-stationarity and we'll Define it in subsequent videos depending upon the level of non-stationarities the order will be integrated of that order and other than that the most important part of that ARA is Arn which we are going to discuss in this video to put it suly an MA process is nothing but a combination of AR and Ma process of different orders say p and Q so if AR process is of order P and M process of order q and we combine them we get an Arma process of order p and Q mathematically the notation is YT as a combination of its own past values these YT minus ones as we did for a process and a combination of white noise processes which is m m minus one and so on so this is our ARA process now as we have already seen we can put both these AR and Ma terms in a more compact notation like this where f l is nothing but 1 - 51 lus 52 L squ and so on the algebraic polinomial similarly the Theta l or the ma process can be written as 1 plus Theta L plus Theta 2 L sare and so on which is the algebraic notation also we said that mes here are white noise processes and these mes the white noise process that these mes are can be defined with expected value of zero their variance which is expected value of M Square as Sigma square and their Auto covariance as zero for all the times which are not same as s for all the t's that are not same as s the auto cence is zero because white noise is a process with no discernable structure what about the ACF and pacf plots interestingly being combination of air and Ma process the ACF pacf plots are sort of combination of AR and Ma process taken separately for example if we put them together both the ACF and pacf plots are exponentially declining in magnitude irrespective of their order for example look at this ARA process of order 1 comma 1 ARA 1 comma 1 because here AR term is order of one and M term is also of order one so this is ARA 1 comma one process and it has exponentially declining structure which is sort of combination of AR and M itself if you recall AR had exponentially declining ACF while single while cut off of pacf same as the order of AR process for ma process it was exponentially declining pacf while ACF was cutting off at exactly the same as number of order of a process if we combine them both for AR and Ma if we combine these terms we get both ACF and pacf which is exponentially declining for both ACF and pacf and that's how you can exactly identify and differentiate between AR me and P Arma model on ACF PF diagrams to summarize Arma processes are nothing but a combination of& ma processes and part of a very wider class of ARA models I in arima would show the level of integration or sort of non-stationarity how What is the characteristic of non-stationarity depending upon the order of non-stationarity whether it is level one level two the iemm would be there but in finance and economic series mostly the I part is modeled separately and then the remaining part is modeled as ARA model so for example if I process I2 first we'll try to remove the non-stationarity of level one or two whatever it is and once you model out nonstationarity you'll focus on Arma part of it in modeling through an expression like this and lastly between AR and ARA process you you can differentiate between them using the ACF and PF plots visually for Arma process both both the ACF and PF are geometrically declined which is different from ACF and pacf for AR and may processes separately in this video we'll introduce to building and estimating ARA AR class of models starting with Arma model the first step is identification of the Arma process that means understanding what is the order of PR and M terms p and Q order of AR process and Order of Q process once you understand the structure and Order of the model then second step is estimation of these parameters for example if it is a araq model you have parameters that is coefficients corresponding to AR and Ma process that you'll try to estimate with process such as OLS and mle OLS is Orly Le Square estimation and ml is maximum likelihood estimate depending upon the context and nature of model for example LS is more suitable to linear modeling while M approach can be used for nonlinear as well as linear then we come to as last step as model Diagnostics that is you test the model generally you try to test the residual of the model that residual should be generally the Assumption with residuals is that they are white noise tum so they should have no Auto coralation structure that is one often the model is also tested for overfitting for example you start with a with an overfitted model and then you test the parameters whether they are significant or not so you keep on eliminating insignificant parameter to arrive at a very parsimonious model the idea behind using a parsimonious model is to choose as less number of AR and Dem terms as possible a parimon model has two very good properties first and foremost it consumes less degrees of freedom which means observations or data observed resulting in lower standard errors so if you have a parsimonious model relative to a bulky and more complex model it will consume risk degrees of freedom and result in lower standard of errors that means good and high power of test the second step is second part is overfitting so a model which is bulky and complex and not less not so much parsimonious may have a problem of overfitting that means it may fit to data specific properties which we often refer to as noise also in physics parland while a more parsimonious model has less tendency of over fitting and it may rather fit to important properties of data which is often referred to as signal in physics parland and which means the implication is that a very bulky model may give very high and good fit with a given sample on which it was trained while in out of sample a new data it may perform poorly in contrast a parsimonious model may be relatively not as efficient in a given set of data in which it was modeled but its relative performance on out of sample data would be extremely good how do we choose between two competing models so there we use something called information criteria for selecting different lags or structure of Arma process that is PQ order ARA orders that is PQ terms we have very three famous AIC K information criteria Schwarz information criteria and also henquin criteria these criterias are nothing but a combination of two terms the formula is different but essentially they are combination of two terms s of trade off or two competing Parts first part this part which is log l which is the log likelihood function sort of log likelihood function which is computed based on the residual errors from the model as you keep on increasing the parameters there are two competing effects at play this residual these residual errors will come down so as you keep on adding more and more parameters this first term will go down as residual errors will go down however the second term which is K increases with parameters so number of parameters so the two competing effects residual this term represents residual for example here this term represents the decrease in decline in residual with addition of new variables while the second term this term captures increase in number of parameters so you try to choose AIC value which is lower K here is K BYT where K is number of parameters small K is number of parameters and T is the sample size log l like I said is the log likelihood of observing the parameters essentially it is computed from the residual squares so here these two competing effects are at play and they their net which is the uh the AI information criterias they are used to compare the models with different structure that is different order of the process Arma PQ process so to summarize in this video we discuss how to build estimate and test an Arma class of model and while deciding the structure of model how do we select the lacks of AR and terms using the information criteria AIC Bic and hqic criteria in this video we'll introduce forecasting with time series models we'll also discuss and compare them with structural models of forecasting and prediction to begin with understanding the time series forecasting process let us understand this notation that is conditional expectation of YT + 1 given information up till time T So this notation expected value of YT + 1 given information this Omega T is for information up till time T it is called conditional expectation of this value Y at a time t + 1 given information available Omega up till time T this is this notation similarly we can write it for time t + 2 t+ 3 and so on now let us think of a zero mean process like white noise process M its conditional expectation at Time t+ 1 given time T is said to be zero for all s greater than zero so for all future periods its forecast the expected value of its forecast is zero there are two very simple methods of forecasting one is called KN no change forecasting this KN forecasting method suggests that forecast of a series Y at time T plus1 given at information time T is nothing but its previous value by T at time T this is also more applicable in the random work process where the process is purely random and therefore the best forecast of tomorrow is whatever the value today so therefore the conditional expectation of this series by at time t + 1 given that the information and values known at time T is nothing but the value at time T itself which is YT this is also called no change forus because we assume that future value the best estimate of future value is to dat's Value itself there is another set of processes which are mean reverting which are expected to revert to their long-term mean and therefore in that case the unconditional expectation is taken and that unconditional expectation of mean of why is taken as forecast sort of long-term mean so so in this case for a process which is mean reverting which is known to revert towards some kind of long-term mean the unconditional long-term mean what is unconditional that we have taken the mean over a long Horizon not specific to a particular information but over a long Horizon that unconditional mean is taken it is also in other words this is some kind of long-term average that is used to forast at any time B now this long-term forecast remain valid for whether you are looking at t+1 or t plus2 or so on and therefore this is unconditional this is unconditional at what time you are looking it's simply the unconditional Focus so we simply write expected value of YT + 1 is equal to that long-term mean or expected value of YT + 2 is equal to that longterm mean now notice there is no conditional this conditional Omega T term is not there which means that this is unconditional now let us compare these time series models and forecasting with time series models with structural models models a structural model would look something like this y tal to beta 1 plus certain explanatory variables like x2t xt3 and so on xkt and their coefficients and then there is this error term this you can think of as a white noise error when you are forecasting or predicting with this kind of structural model which is obviously driven by some kind of theoretical underpinning the conditional expectation of YT at time t with given information at T minus one would appear something like this however this kind of forecasting and because these betas are constant they are sort of population parameters so they are taken out so inside expectations operator you have taken expectations on both Sid we have x2t x3t these are the values in the expectation now in order to conduct the forecasting or prediction or what we call is conditional expectation of YT we need forecast of x2t x3t xkt which are time series processes themselves and that makes this prediction with the help of structural model extremely complex process because now you have to rely on all those K variables and do the modeling and prediction for these variables themselves this is a difficult part and therefore one may have to examine the historical values of all these XS maybe find their some kind of mean or expected values first you have to find the expected values and then in turn you have to forecast the expected value of it however rather than going through this cumbersome exercise if you are relying on simple time series model of YT by modeling through its own values like YT minus1 BYT minus 2 and the white noise information shocks Ms that have arrived currently and previously you can very easily model BYT without considering these values the idea here is that this structural relationship or any other kind of structural relationship will remain any way valid and therefore this relationship will remain also valid historically that means for YT minus one and YT minus 2 and therefore when we are modeling YT with YT minus 1 YT minus 2 and so on essentially we are factoring these relationship that have appeared or captured their information historically so this relationship the information pertaining to such structural relationship is in historical terms is already captured in YT minus1 YT minus 2 and therefore when I'm modeling YT with its own historical values and white noise error terms such as this throw some kind of ARA process I'm essentially without even going into this structureal model I factoring this and therefore this kind of modeling is extremely powerful that it saves a lot of time and effort in modeling the structural model and able to capture the information content of this struct process itself so to summarize this video we have understood what is time series forecasting what are the elements of Time series forecasting and how it is a better way when we are doing some kind of future forecast or prediction as compared to structural models it while it takes the information content of those structural models is still it saves or avoids lot of problems in terms of information collection and GA information gathering and building comers some models as done with the structure models in a series of next few videos we'll try to understand forecasting process with ARA class of models in this particular video we'll understand forecasting with ma model let us examine this simple ARA PQ model at time t with steps into future the forecast would appear something like this notation is Ft plus s representing at time t s side forecast now it will have two components first AR component that is its own previous terms if the actual value is not available then forecast F will be employed which is the previous values of the series itself and its coefficient AIS which are Auto regressive coefficients and then you have ma terms White Noise process mes his previous values and current values of MW with m ma coefficients so let's start with forecasting of ma process let us take a simple example of ma3 order process YT equal to Mu this is the constant term which we assume some kind of long-term constant value which is known to us from historical experience and then three terms Mt minus 1 mtus 2 mtus 3 and their coefficient Theta 1 theta2 theta 3 and the current information or innovation shock M we'll assume parameter constancy that this Theta 1 Theta 2 and Theta T will remain unchanged as we move ahead into future so if this relationship holds the one step ahead for cast YT + 1 will be obtained by simply just adding one in this subscript so the model will become mu plus Theta 1 Mt plus Theta 2 mt- 1 and so on up till theta 3 Mt minus 2 Plus Mt + 1 in this model we can obtain the forecast by simply taking expectation which is the forecast essentially nothing but conditional expectation of YT + 1 using information available up till T that is conditional up till Sigma T so you'll apply the expectation operator and the constant terms such as Mu and Theta 1 Theta 2 theta 3 will remain unchanged and will be taken out however one important Point till now we have been saying that the conditional expectation of error term as zero because these are white noise processes however when we are standing ahead in today all the information or these information shocks what we are if you recall we said these white noise processes are nothing but being white noise processor are used to model the information shock that has arrived in the process so whatever information shocks have arrived till date they are already known so their conditional expectation is not zero we'll not take it as zero but we'll use the actual values so for example up till information T up till time T we have all the information available so for that we'll use the actual observed values of M M minus1 M minus 2 however for MW +1 because this is not known its expectation obviously will be zero so expectation of this Mt + one will be zero but the conditional expectation of these terms Mt Mt minus1 mwus 2 will not be considered their actual values will be taken so in this Spirit when we take expectations this M plus one will become zero but for all the other terms we'll use the MU plus Theta 1 Mt plus Theta 2 mtus 1 plus theta 3 mtus 2 with the caveat that these Mt mtus1 Mt minus 2 are actual values that are already observed standing time T so with this being the case we already have the forecast of FT1 one step ahead in future which is this now let us do the forecasting for Two Step Ahead in future which is Ft comma 2 which is nothing but conditional expectation of YT + 2 again using the information up till time P now in this case we need to add the subscript here plus one so for each we will add one to subscript and then we recall that when the expectation operator is applied t + 1 will be zero but for Mt and Mt minus one actual values will be employed and therefore this term will get eliminated and we are only left with mu which is the constant term plus Theta 2 Mt plus theta 3 Mt minus one which is our forecast for Two Step Ahead in a very similar fashion for three step at forast we'll take the conditional expectation of YT + 3 given information Omega T so we'll add another plus one in the subscript here M plus 2 m+ one this will become M itself now when we take this expectation again please note this Mt plus2 is in future so it conditional expectation is that is expectation of Mt + 2 using information up till Sigma T Omega T it is zero so this will be zero this will be zero because they are in future only for Mt the actual value will be employed and therefore we are left with mu Plus theta3 Mt so this is our Focus for three now please recall this is a M3 order process order of three so only only information that is available up till three step ahead will be used after that there are no mes that are actually available so now all the terms will be in future that is me + 1 m + 2 and so on case in point four step ahead forast which is nothing but conditional forecast of YT +4 given Omega T now in this case as soon as we had one step ahead we have t+ 3 T plus2 and t+1 all these white noise MW are in future so their conditional their expectations conditional expectations given Omega T which is the information available till T will be zero so all these terms will be zero and we are only left with mu and I hope you can see where we are going with this all future forecast T comma 5 T comma 6 and so on will become mu only which is the constant or some kind of long-term mean of this process as we said earlier for practical implications we'll have this mu as zero so in that case this if if this mu is taken as zero all these values will become zero and only those values up till three forecast and why only three because this is a this is an MA three order process since the M3 order process has a memory of only three periods any forecast of four or more periods will converge to that long-term un conditional mean mu or The Intercept term if this is being taken as zero then this part will become zero only and only up till three period Focus Will Survive to summarize in this video we have discussed how to focus with Ma process we noted that Ma forecast become or converge to some kind of long-term conditional mean for any term higher than the order of process so if it is an maq process any forecast which is more than Q steps maybe Q + 1 or Q + 2 will converge to the long-term unconditional mean which is The Intercept term if it is zero then that forecast will be zero and only those forecast that are up till Q steps side in future will survive for an maq order process in this video we'll try to understand forecasting for AR processes let us consider a simple ar2 process its structure would be YT equal to Mu which is again the constant term maybe taken a zero also 51 into YT minus1 and 52 into Y tus2 which are the previous values along with the coefficients Auto regressive coefficients 5152 and the white noise error term now as we'll see shortly very shortly that unlike ma process which was restricted to the number of lags of white noise terms a process has a infinite memory we'll see that so let's understand what will be the t+1 forecast here so forecast at t+1 would be the constant term plus 51 and we'll add for each of these subscript We'll add one so here we have M YT YT YT minus one and then this m+ one which is the White Noise error in this case when we are Computing the forast for one step ahead and by taking the conditional expectation of YT +1 at information available up till time T please recall that up till time T we have YT and YT minus one already observed so we'll not be taking their expectation but actual values because they can be observed and when we are Computing the expected value of this Mt + 1 because this is not observed it will become zero so our expectation operator will operate on this mu YT YT minus 1 and so on but their actual values are available so they will be taken but but when we compute the expected value of this this will will become zero so we are left with mu + 51 YT 52 YT minus 1 these are actual values that are observed already because we are standing at time T in this fashion we have obtained the forecast One Step ahe forecast like this we can similarly proceed for step two two step forecast by ft comma 2 which is nothing but conditional expectation of YT +2 at time T information available pre time T again we'll add one to the subscript so we'll get YT + 1 here YT here and Mt +2 here now while applying the expectations operator please note that mu is constant so it will remain unaffected YT + 1 now is not observed at time T so we have to take its expected conditional expectation which we have already computed as ft comma 1 YT is already observed so it will remain as it is so we'll have mu which is the same with 51 we'll have the expectation of it + 1 which is Ft comma 1 which is nothing but conditional expectation of YT + 1 given information available up till time t plus 52 into YT because YT is actually observed now I hope we can see where we are going with this if we compute the third step Focus which is the conditional expectation at time t+ 3 omega T we add another plus one to the subscript so it becomes YT + 2 YT + 1 and Mt + 3 the expectations operator will give us mu here but now YT + 2 will be nothing but ft comma 2 which is computed in the previous step and ft comma 1 which was completed in this step and therefore our resulting forast our resulting forast which appears like this we can obtain a more generic expression look at this ft comma 3 plus mu into 51 * ft comma 2+ 52 * ft comma 1 so the generic forecast will appear like this which is nothing but ft comma s step side forecast with a constant term immediate previous term s minus one and next previous term s-2 why these two terms precisely because this was an A2 process so that's why we are having these two terms but with these we can go ahead infinitely in future and we'll always get a forecast which is not same as Mu or zero as was in the case of ma which means for every n x steps in future we can obtain some kind of conditional forecast using previous two values which do not become zero unlike the case was ma in MA as soon as we are ahead of the ma order of the process which was maq the forecast were converging to Mu which is the conditional mean or zero if in case it was Zero but here every time we observe some kind of forecast depending upon its immediate previous values and that is why we said this AR process has infinite memory more importantly now that we have forecast for this ARP order kind of process and maq as we seen earlier we can combine them and obtain the complete forecast for araq process to summarize this video we have understood how to for for AR process we noted that unlike ma process which were restricted to the order of the process the forast for ma process the focus for air process have infinite memory that that they can be made infinitely long Horizons but in conclusion if you want to forecast for araq model you can separately forecast for AR and Ma processes and then add them together to get the complete full Arma PQ order forecast in this video we'll be introduced to some basic terminology Rel to forecasting models first we'll start with in Sample versus out of sample forecasting for example let's say we have data available from January 1990 up to December 199 one approach to develop a model is to use the complete data from January 19 to December 1999 and test its Fitness or try to see how fitted values will appear but this is not a very good approach because generally it is expected that when we are training the data using this data training the model the model would generally do well on this data itself so a more evolved way is to take a certain set of the sample maybe January 1990 to December 1998 call it training data set part of the sample on which dat the model is trained and parameter estimated and a certain set of sample which is left out or hold out sample from January 99 to December 99 these set of observations are not used in as a part of train data set and these observations are used for testing the model so we use certain set and this is up to researcher how many generally a good ratio is 70 30 or 8020 so on 70 to 80% of the observations the model parameters are trained and once the model is trained we estimate we try to predict on the set of observations and then test the model accurate which is called out of sample forecasting so model this was the in Sample estimation and this is out of sample forecasting or evaluation of the model period the next is what is the forecasting requirement for example I may want to forast for one step ahead somebody may want to forus Two Step Ahead or three step ahead generally it is Will accepted that if you go further and further if you go s steps in future where s is pretty large you're forecasting it accuracy decreases so as a researcher I should have some idea some experience about my data that whether one step or two step or three steps in future forecast are more efficient rather than a very lengthy forecast third is rolling versus recursive forecasting let's understand it so let's say I'm planning for a one two or some kind of step and forecast a rolling window approach would start with a certain initial set maybe January 1990 to 1998 December and then forecast maybe one two three steps in future now when as a researcher maybe I have found that as steps maybe three steps in future three step side forecast is good enough for me so when I'm going to the next three steps there are two ways either I can keep my initial window fixed and add the next three steps one t + 1 t + 2 and t + 3 in my current set of observations and then make use of original sample s observations plus these three newly added observations to make the next set of foras another approach would be to rather than than keeping the initial this is called the previous approach was called recursive window because the initial period was kept fixed and as we move ahead in future we keep on adding each observation one by one we keep on adding to our original s original sample so our sample increases the second approach is to have a rolling window approach where as we keep ahead in future we add one observation but at the same time we exclude the last observation so this January 1990 will be excluded when January 99 will be added when February 99 February will be excluded and February 1999 will be added and as we keep on moving ahead our sample estimation period will remain fixed that is approximately this length is 180 month that remains fixed as we keep on moving ahead so we exclude one period when we add one period This is called rolling window approach both these approaches have a trade-off for example with recursive window the benefit is that your sample estimation period in Sample period becomes larger but at the same time the previous values that are still not taken away from the sample they may not be in sync with the current market conditions they may be probably more stale and therefore some of the very old values may not be up to the date and may not give me correct picture while with rolling window we limit our sample size to in this case 180 or a certain number but at the same time as we move ahead we keep on excluding historical values so that our sample that on which we are training the model is more in sync with the current market conditions so to summarize this video we learned some terminologies related to forecasting these included One Step versus multi-step forecast we have in Sample versus out of sample prediction and rolling versus recursive method of forecasting in this video we'll try to understand how to determine the accuracy of forecasting as we noted earlier that often it is important to have model trained on in sample on a given sample data while use the hold out sample or out of sample the second cut of sample for out of sample prediction and then find the accuracy of forecasting on this out of sample prediction set let us see and proceed how to do that let's say you have built some kind of armara model and you have as using the method that we have already seen you have forecasted certain values for 1 2 3 4 and five steps ahead so these are your fasted values on out of sample because this is out of sample you already have the sample values available actual sample values to test the approach to do is compute some kind of error measure in this particular example we'll use use mean square error mean absolute error and root mean square error First Step we'll compute the difference the difference between focussed and actual values which is the error but generally there are two ways either you take the square or the absolute value why we are doing this because if we are using some kind of summative measure summation for positive negative values they will cancel each other out so large positive negative errors will cancel other oute and my measure will not be good so generally a good measure is like mean square mean absolute they tend to take either square or absolute values of Errors so once you have the square of Errors you take the difference which is your error so your error will be and then you square it you'll get those values and then you can sum them up and take the mean how to do that so for example these are our squared values these are square errors this is the squared number once we have the summation of this Square I can divide it by five to get what we call as mean square error Ms squares and then divide to compute mean and this is mean square error also a more advanced version of this mean square error is root mean square error which is nothing but the square root of this msse in this case 08 I can take the square root to get the0 28 the second way is to take the absolute value so again I can compute the error F minus a for C minus actual take its mod or absolute value and then I can sum up all these values and divide by number of observations for example in this case I have these absolute errors I can sum them up all five and divide by five to get what we call as mean absolute error to summarize in this video we have understood how to compare across different models by using some kind of error measures these error measures like MSC ma RMC we have seen there are more possible err measures but please note they do not have any meaning in absolute Sense on it on their own they do not have much meaning per se when they are more useful in comparing across competing models and why they are being used because we want to compare different models on their capacity of out of sample prediction so using the actual and forecasted values from different models we compute these error measures on the out of sample or hold out sample part of the data and then compare across different Computing models to compare their error values on its own these error values do not make much sense we do not have a way to say that this is bad or good only when we have competing models and their respective errors we can compare them across different error measures also what kind of error measures are being used that is up to researcher in different cases different measure are used for example when you are working with financial market returns probably the positive errors are not so much as bad as negative errors so an investor would rather like to have those models where or evaluate a model based on its negative error how much extremely negative returns or losses he is witnessing maybe he's not very worried about positive returns or gains he probably is more interested on the negative side negative tail of Errors so what me error measures we are going to use that is up to the researcher and the context but these errors will be used on out of sample forecasting or prediction in comparing across different Computing models to summarize an ARA process is simply a combination of AR and Ma processes it can be easily identified on on ACF and pacf plots as exponentially declining process on both the plots to build an Arma model first we identify the order of the process next we estimate the parameters through some kind of estimation procedure such as Os or mle procedure lastly we diagnose the model with the help of residual Diagnostics and out of sample prediction accuracy of the model for lack selection process we use information criteria one of the important objective here is to find extremely parsimonious model which fits the data well in a strict theoretical sense for an ideal model the residuals extracted from the model should not have any discernable structure and should act as a white noise process forecasting with herma model requires conditional forecast for next one to S steps ahead for those terms where information is available the actual values are employed where the actual values are not available conditional expectations are employed often models appear to be extremely efficient when examined with the sample on which it was trained this often leads to overfitting process that is a very bulky and complex model which does not do very well in new data that is why different Computing models are tested on out of sample or hold out sample which was not used while training the data the two key terminologies in forecasting are first rolling sample forecast which employs fixed window of sample which is moving in a step-by-step manner second is recursive window for cast where initial point is fixed and new data points are added and Sample increases with each step in the forecasting process the choice between the two methods is a trade-off between a sample which is a large larger in size versus that which is in sync with the current market conditions lastly across different competing models we compare their performance on out of sample forecasting accuracy through error measures such as msse which is mean square error ma mean absolute error or rmse for example which is root mean square error and then select the best [Music] [Music] model [Music]