**Introduction**

Transfer function model is a unidirectional relationship between input and output.

**Problem/Hypothesis of the time series data:**

To analyze the relationship between Consumer Price Index (CPI) the Gross Domestic Product (GDP) of the Indian market.

**Number of records in the dataset:**

The dataset contains the history of CPI and the GDP for the past 60 years of the Indian market (one entry for each year).

Data source: https://www.data.gov.in/

**Transfer Function:**

On our initial visual analysis, we could see that both the input and output parameters are of different scales, the GDP was in lakhs of rupees and the CPI is in percentage, so we have transformed the dependent variable (GPD) with log_{10}.

The relation between the input and output is mostly linear with some discrepancy due to white noise and the auto correlation between the output variable. On business terms, we understand that the Consumer Price Index and GDP of a India were related to each other with positive correlation and having a good interaction. This can be confirmed by the given scatterplot matrix below:

With the above understanding we continue to the transfer function analysis for the above dataset.

With initial Time Series Basic Diagnostics, we see that the for the output variable the ACF and PACF both are tailing off; hence this is an ARIMA model. But, we see that the data is already stationary so we have not done any differencing. Similar pattern is followed by the input series (CPI) also.

For Pre-whitening, we could see that the auto correlation is significant after lag 14, i.e nearly after 15 years. By Indian Government standards, the budget is revised every 5 years and due to the slow growth of economy the impact of CPI on GDP is seen after 15 years. But it is very prominent during the next 5-7 years as seen in the below ACF graph.

For the transfer function, we expect that B^{15} and other variables are to be prominent. The above analysis resulted in the below transfer function wherein the relation between the output variables and the input is seen to be prominent. Thereby confirming the delayed impact of CPI after 15 years on the GDP of India.

Since, the pre-whitening process gave us a model of (2,0,2) and we could also see that the peak in pre-whitening lag plot is at 19 and the significant values start from 15. So s1 is given a value of 4. Below is the transfer function model summary.

The actual transfer function which depicts the current GDP is impacted by the 5 CPI observations that are nearly 15 years old. This also gives us an idea that the 5-year budgeting is having a prominent impact on the GDP.

**Conclusion**

From the above analysis we have concluded that the GDP estimate on the input variable (CPI) is as given below:

This gives us a clear understanding that the current GDP is dependent on the 3 previous years’ GDP and 6 years of CPI (independent variable) which is 15 years apart. We also notice that as the year grows, the older years are having minimal impact.