A statistical Study of Price of Gold

Gold has long been considered one of the most precious metals, and its value has been used as the standard for many currencies (known as the gold standard) in history. Ancient India accumulated gold through trade of Indian spices with Roman Empire and by gold mining operations, predominantly in the regions of Hatti and Kolar of modern-day Karnataka. 1. There has been drastic increase in the prices of gold since 2001. Gold prices have been increased by 90% during last 10 years. 2


INTRODUCTION
We started with the literature review of the previous researches on the effect of inflation, silver price, USA dollar trade, personal disposable income, crude oil price, interest rate with gold price.The theoretical or conceptual framework was presented at the following part, identified the network relationships between the variables.

EXCHANGE RATE:
An exchange rate is the price of nation's currency in terms of another currency.As a rule, when the value of the dollar increases relative to other currencies around the world, the price of gold tends to fall in U.S. dollar terms, it is because gold becomes more expensive in other currencies.Exchange value of US Dollar is an important factor in determining the gold price in India.The appreciation or depreciation of US dollar creates fluctuations in gold price.• Email: editor@ijfmr.comIJFMR23069249 Volume 5, Issue 6, November-December 2023 2

CONSUMER PRICE INDEX (INFLATION)
Inflation is defined as a sustained increase in the price of goods and services.Over time, it erodes the value of a nation's currency.In other words, it erodes purchasing power.According to empirical findings, highly positive correlation is found between Gold Prices and CPI rate of our country; however, there is a short-term relationship between inflation and gold prices.
Index is a statistical aggregate that measures change.BSE SENSEX is the market weighted stock index of top 30 companies. .Sensex is composed of 30 of the largest and most actively traded stocks on the BSE, providing an accurate gauge of the overall growth, development of particular industries, and booms and busts of the Indian economy.There is an inverse relationship between Gold Prices and Sensex.

PERSONAL DISPOSABLE INCOME:
Personal Disposable Income (PDI) is the amount of money that households have available for spending and saving after income taxes have been accounted for.According to a report by the World Gold Council, annual data from 1990 to 2015, gold demand is seen to rise with income levels.For a 1 percent increase in income per capita gold demand rises by 1 percent and for a 1 percent increase in prices, the demand for gold deters by 0.5 percent..Therefore, we can conclude that a higher demand will drive up gold prices by a higher margin.

INTEREST RATES:
Generally, real interest rates are negatively correlated with the price of gold.i.e. it is relatively expensive to hold gold in the portfolio when real interest rates are high, and relatively cheap when real interest rates are low.Thus, the higher the interest rates are, the higher are carrying costs.However, the relationship is not linear.Gold prices tend to increase significantly only during the periods of negative real interest rates.

CRUDE OIL PRICES :
Crude oil is among the eight core industries in India and plays a major role in influencing decision making for all the other important sections of the economy.Crude oil prices can be used as a reliable proxy of movements in gold prices since the two have a strong direct relationship.This is because, like oil, gold too is mined from the ground, is standardized and interchangeable.Due to the fact that, energy is the dominant production cost for gold.

SILVER PRICES:
Gold and Silver prices are correlated to other commodity prices like, global oil benchmark, global inflation.Silver prices appreciate with trade and growth imbalances against the US.Silver prices rise with falling real interestrates.Prices of both, rise together with economic improvements in emerging markets.Thus, silver and gold prices establish a direct relationship.
Thus, these 7 factors mainly affect the prices of gold.

Data collection
For studying about this topic particularly for India, we took help of the government sites, we referred secondary data.We have collected this secondary type of data from the website of Reserve Bank of India (RBI).The Reserve Bank of India is India's central bank which controls the issue and supply of the Indian rupee.RBI is the regulator of entire banking in India.We have used Handbook Of Statistics (HBS) which disseminates time series data on various economic and financial indicators relating to the Indian Economy.

CONCLUSION:
There is direct relation between gold and exchange rate.

II. GOLD PRICE VS CRUDE OIL PRICES MODEL:Yg=β0+β1*Xcr
Where, Yg=gold price Xcr=crude oil prices HYPOTHESIS DEVLOPMENT: HO: β1=0(there is no relation between gold price and crude oil prices).H1: β1≠0(there is relation between gold price andcrude oil prices).TEST CRITERIA: Reject Ho if p-value is less than 0.05 or t-statistic is larger than p-value.accept otherwise.

DECISION:
Here, p-value is 0.03503 <0.05.reject H0, Hence accept H1.CONCLUSION: There is relation between gold price and crude oil prices.

Artificial Neural Network (ANN)
Estimation is a process that produces a relationship using examples that have definite consequences and allows you to comment on the new situation through this derivation.As a prediction method, Multi-Layer Neural Network (MLNN), which is an artificial neural network model, was used in this study.1.The network architecture has an input layer, hidden layer (there can be more than 1) and the output layer.It is also called MLP (Multi Layer Perceptron) because of the multiple layers.2. The hidden layer can be seen as a "distillation layer" that distills some of the important patterns from the inputs and passes it onto the next layer to see.It makes the network faster and efficient by identifying only the important information from the inputs leaving out the redundant information.
In the study, the MLNN model was used to estimate the gold prices.As input to the model, data sets consisting of input variables (exchange rate, interest rate, CPI, PDI, silver prices, crude oil prices, BSE) were used.In this model, 8 neurons in 2 hidden layers.In this method program attaches weights to different nodes in hidden layer and using back propagation algorithm and predicts the required values of dependent random variable.

IME SERIES ANALYSIS
A time series is a series of data points indexed or listed in time order.It can be continuous trace or discrete set of observations.Here we are dealing with observations taken at discrete time periods.By appropriate given relatively more weights in forecasting than the older observations.It is usually used to make short term forecasts.
Here our data shows trend as well as seasonality hence we use triple exponential smoothing.
The basic equations for this method are Where y is the observation S is the smoothed observation b is the trend factor I is the seasonal index F is the forecast at m periods ahead t is the index denoting a time period Here we have used monthly data of gold prices.This method is mostly useful in case of short term forecasting.Hence analysis will be done on the data from Jan 2005 to July 2010 and data of next six months will be used for validation of analysis.Output: Now we check the normality of residuals.Normality of residuals is an assumption of running a linear model.If the residuals are normal means predictions are valid.

OUTPUT: Shapiro-Wilk normality test
As p value is greater than 0.05 we can say that residuals are normally distributed.Now we check the randomness of residuals by run test.If the residuals are random then it's just the white noise.The Run test procedure is as follows: We take median and compare it with each value.The values below median are put as 0 and above values are put as 1.We convert it into vector and run the test.

p-value = 0.09923
As the p value is greater than 0.05 we can say that residuals are random.r=r-median(r) q=c(0,1,1,1,0,1,1,0,0,1,1,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,0,1,0,1,1,1,0,0,1,1,0,0,0,0,0,0,1,1,0,0,0,0, 0,1,1,1,1) q=as.factor(q)runs.test(q)R code for checking normality of residuals: res=tss.dec$randomplot(res) shapiro.test(res)ARIMA MODELLING ARIMA stands for Autoregressive Integrated Moving Average.This model is fitted to the time series data either to better understand the data or to predict future points in the series.The AR part of ARIMA indicates that the evolving variable of interest is regressed on its own lagged values.The MA part indicates that the regression error is actually a linear combination of error terms whose values occurred contemporaneously and at various times in the past.Non seasonal ARIMA Models are generally denoted by ARIMA(p,d,q) where parameters p,d and q are non negative integers.P= order of autoregressive model q= order of the moving average model d= degree of differencing Differencing: In statistics, differencing is the transformation applied to the time series data in order to make it stationary.In order to difference the data, the difference between consecutive observations is computed.Differencing removes the changes in the level of time series, eliminating trend and seasonality and consequently stabilizing the mean of the time series.
Analysis will be done on the data from January 2000 to December 2018 and data of next six months will be used for validation of analysis.Here first we convert our data into time series data and we check its stationarity by adf test and kpss test

Conclusion:
1. From the graph of decomposed data, we could infer that our time series consists of seasonality, trend and irregular variations.2. In Holt Winters forecasting, the random or error component of series is normally distributed and completely random.Hence it is a white noise indicating good prediction.3. Our original time series is non stationary and we convert it to stationary using differencing method on log values.4. For ARIMA Modelling, the actual and forecasted values have very small difference.5.For Holt Winters Forecasting, the actual values lie perfectly within given confidence Intervals indicating good prediction.

CONCLUSION
We have made predictions using three techniques time series, stochastic and neural networks.There are some advantages and disadvantages of each technique.
As a primary tool of data analysis we used simple linear regression to analyze the relationship between dependent variable and independent variables.
Due to wide range and different units of dependent variables we couldn't get appropriate predictions by regression.This problem is tackled by method of neural networks.There is no specific method to determine the parameters of ANN model.So by trial and error method we have to find out best parameters that suit our data.Though it is tedious process, outcomes are worth as model learns pattern by itself and predict future forecast.
Method of Neural Networks considers all independent variables.But it is not always feasible to obtain data for each independent variable.Hence by only using dependent variable data we can do forecast by time series.
There are many fluctuations in gold prices due to trend and seasonal factor.Due to presence of seasonality we have used triple exponential smoothing.By this method we can do short term forecasting.Long term predictions do not meet accuracy Hence we use one other method that is ARIMA Modeling.In all the techniques , predictions using neural networks were best.By keeping an eye on daily prices of Gold , one can decide whether to invest in Gold or Stocks.It is clear from exploratory analysis that gold prices have increased between year 2000-2019 by nearly 15000.It is observed that in respect with increase in rate of inflation, there was also rise in price of gold as it is observed that price of dollar affects gold price directly.
11 12 13 14 15 16 17 18 19 Gold Prices (in rupees per 0.1 gms) exchange rate (per U.S dollar) GOLD PRICES VS CRUDE OIL PRICES 11 12 13 14 15 16 17 18 19 Gold Prices (in rupees per 0.1 gms) Crude Oil Prices (in hundred rupees per barrel) III.GOLD PRICE VS BSE MODEL: Yg = β0 + β1 * Xb Where, Yg : Gold Prices Xb : BSE Rate HYPOTHESIS DEVELOPMENT: H0 : No relation between gold prices and BSE rate i.e, B1 = 0. H1 : There is a relation between gold prices and BSE rate.TEST CRITERIA: If p-value > 0.05 Accept H0.DECISION: Here, p-value = 0.001211 < 0.05 Reject H0.CONCLUSION: There is a relation between gold prices and BSE rate.IV.GOLD PRICE VS SILVER PRICE MODEL: Yg = β0 + β1 * Ys To Test : H0 : No relation between gold prices and silver prices i.eβ1 = 0. H1 : There is a relation between gold prices and silver prices.TEST CRITERION : If p-value > 0.05 Accept H0.DECISION : Here , p-value = 0.000950 < 0.05 Reject H0.CONCLUSION : There is a relation between gold prices and silver prices.
12 13 14 15 16 17 18 19 Gold Prices (in rupees per 0.1 gms) silver prices (tss);p; Decompose Function: the decompose () function is used to decompose the given time series into seasonal, trend and irregular components using moving averages.the function first determines the trend component from data and removes it from series then seasonal figure is computed by averaging.the seasonal figure is then centered.finally the irregular or error component is determined by removing trend and seasonal figure From the graph, we can infer that trend , seasonality is present in the data.

Data from year 2000 to 2019 used for Regression Analysis and Artificial Neural Network Modeling year
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Monthly Data From Year 2000 To 2019 Used For Time Series And Stochastic Process Analysis
Reject Ho if p-value is less than 0.05 or t-statistic is larger than p-value .acceptotherwise.