GARCH estimation using maximum likelihood

In my previous post I presented my findings from my finance project under the guidance of Dr Susan Thomas. The results in my paper suggested that there are macroeconomic variables, particularly the INR/USD exchange rates, that help us understand the dynamics of stock returns. Although the results that I obtained were significant at 5% level, the weight of my assertion needs some more robust check before we consider the matter closed. One possible source of discrepancy we identified was that the error terms could be heteroskedastic. Meaning that one of our assumption of classical linear regression model (CLRM) estimation viz. homoskedasticity is violated. The resultant coefficient estimated, in case of heteroskedasticity of the error terms, can have underestimated standard errors, which in turn might lead to false acceptance/rejection of our null hypothesis. There are several parametric as well as non-parametric ways to remove the effect of heteroskedasticity in the error terms on the coefficient estimates. However, the method that I adopted to correct for this effect in my model was the path breaking conditional heteroskedasticity modelling propagated by Robert Engle for which he was awarded the Nobel prize in 2003.

The idea behind auto regressive conditional heteroskedasticity (ARCH) model is quite simple and straightforward. One needs to model the mean equation (this is your regression model) and along with that there has to be a specification for the modelling the squares of the errors. Suppose we have a simple AR regression:
     
z_t = φ*z_t-1 + e_t        (Mean equation)

Now to model the conditional variance of the error terms all we need to do is a simultaneous estimation of the squares of the error terms in the following manner:

e_t = ѵ_th_t^1/2 ,  where ѵ_t  ~ NID(0,1)

h_t = α₀ + α₁e_t-1²      (Variance equation)

The above specification of the mean and the variance equations is termed as a AR(1) ARCH(1) specification. This simultaneous estimation takes into account the particular form of heteroskedasticity (ARCH(1) in this case) and estimates the 'φ' coefficient accordingly. Now the above simple specification tends to pose another problem of lag selection, what lag should be considered for the variance equation. To deal with this and several other shortcomings of the simple ARCH model, Bollerslev(1986)  proposed a generalized ARCH model (GARCH). The only difference being that the variance equation now becomes:

h_t = α₀ + α₁e_t-1² + βh_t-1

Which is nothing but a GARCH(1,1) model. The beauty of this specification is that a GARCH(1,1) model can be expressed as an ARCH(∞) model. For those who are interested in learning more about ARCH and GARCH processes and the mathematics behind them here are Dr Krishnan's notes that provide an in-depth understanding on the matter. The reason why the ARCH and GARCH models rose to such prominence was because they offered us a method of not just correcting the standard errors of the estimates (as like other robust correction methods) but also provided a functional form for modelling the variance. This idea of modelling variance was heavily adopted by financial researchers. Considering the standard deviation (or variance) of asset prices as a crude estimate of volatility, financial researchers used ARCH and GARCH models to model asset price and/or other high frequency index volatility.   

Now that the idea behind the GARCH modelling is clear let us plunge right into writing the conditional maximum likelihood function for a GARCH process.

Utkarsh was the generous one who provided me with the basic structure of the codes that I then customized to solve this problem at hand. Well now let me confess that a similar result or in fact a more accurate estimates of the coefficients can be obtained using unconditional maximum likelihood estimates that are offered by any of the high priced computer packages like EVIEWS, RATS, SAS etc but my motivation (apart from learning ofcourse) was to figure out a way that could give me the flexibility of modelling any functional form of heteroskedasticity. The above codes have been presented for the estimation of a GARCH(1,1) model but I could do a simple manipulation in the definition of h_t to fit any arbitrary functional form. 

P.S Sincere thanks to Utkarsh for letting me share the codes. Any feedback/suggestions are welcome.

Revisiting basic macroeconomics : Illustrations with R

Prologue
After 3 semesters of studying economics at IGIDR, the basics of macroeconomics still elude me. What policy shifts what curve? What determines the slope of IS-LM and AD-AS curves? What exactly was Keynes contribution to Economics? How do all these curves move in tandem? Well, I would not be surprised if you are a graduate student in economics and face similar doubts with these basic fundamentals. So in an attempt to demystify or rather simplify these macro fundamentals for econ (as well an non econ) students Isha and I have attempted this brave move of presenting the fundamentals of macroeconomics in a concise manner. We will try to be as explicit as we can but we might end up assuming some prior knowledge of economics. In such cases, we would encourage the readers active participation in improving the quality of the post wherever he/she thinks and idea is expressed with inadequate explanation.

The story
Any theory/idea, be it in pure science or economics, emanates with a hypothesis that is supported by some assumptions. As the study on the subject progresses researchers try and play with the assumptions and see if they can make it better in terms of explaining the real world as we see it. Macroeconomics is no exception, classical macroeconomics dates back to the 19th century with major proponents being Adam Smith, David Ricardo, Malthus, John Stuart Mill, J.B Say, etc. They believed in the functioning of free markets, that is a decentralized mechanism to ensure demand meets supply would result in efficient allocation of resources. Any counter-cyclical measure by a centralized government in order to smoothen the fluctuations in the business cycles would be self defeating. The centralized government, according to these classicals, was primarily to keep the budget deficit (difference between the govt. spending and tax revenue) in check. Meaning that in times of recession, when there is less revenue being generated by taxes, the government should cut spending in order to keep the deficit in check. The reason for them to propagate this idea was the if govt. has to increase spending, in times of recession, it would lead to a rise in the interest rates (since it has to borrow from the citizens) and this increase in interest rates would crowd out private investment (which is what we don't want at times of recession). This ideology was called into question during the great depression of 1929 in the US. This is when the "Say's law", which was at the heart of classical economics, failed to hold. There was mass unemployment, which was invoulantary (as opposed to the classicals view of only voluntary unemployment), insufficient demand for the produced goods, in short "Great" was an adequate adjective to describe the depressed economic scenario during the 1930's.

The Great depression of 1929 provided researchers with the much needed opportunity to revise/update the assumption that they were playing with. J.M Keynes made a break through contribution in revising the assumptions and ideologies of the classicals. He argued that market left completely to themselves might not lead to efficient allocation of resources. This was a paradigm shift in the way policy makers and researchers thought about macroeconomics. He argued that in times of recession the govt. should boost spending in order to revive the economy, contrary to the idea that the classicals propagated. Government spending and private investment were seen as complements and not substitutes. Private spending was determined not only by the interest rates in the economy but also by the expectation of future profitability (famously called the "animal spirit"). Another set of assumptions that he relaxed from the classical framework was the full flexibility of wages and prices. Classical economists believed that the market mechanism ensured that the prices of commodities adjusted instantaneously and fully to make the supply = demand. For eg. if there are 10 apples being produced and the demand turns out for only 8 apples, classical economists argues that the price of the apples would adjust in a manner (fall in this case) so that the demand rises instantaneously and exactly by 2 apples and the equilibrium is maintained. Keynes on the other hand argued that due to rigidities in the market (sticky prices, labor unions, menu costs, etc.) the price adjustment would not be instantaneous and hence the additional demand, in the short run will have to be created by the govt. by spending more. Either it can pitch in and buy the additional 2 apples or it can expedite investment in an avenue that absorbs the labor retrenched due to insufficient demand in the apple market.

Enough of literature, now let us get our hands dirty with some basic maths and visualization of some economic concept that would help us appreciate the above ideologies. We would start by illustrating the difference between the 2 idelogies using the IS-LM  framework (also called the Investment Savings/Liquidity preference Money supply). The basic national income identity finds mention in the first chapters of most of the introductory Macroeconomics textbooks hence we shall start by the same identity:

Derivation


For complete and comprehensive proofs of the above equations you can refer to a text book by William Branson or another textbook by Dornbusch and Fischer.

IS curve: The points on the IS curve represent the combinations of rate of interest (i) and output (Y) for which the goods market are in equilibrium. Meaning, at these combinations of "I" and "Y", the aggregate supply of goods equals aggregate demand for goods in the economy.

LM curve: The points on the LM curve represent the combinations of rate of interest (i) and output for which the money market is in equilibrium. Meaning, at these combinations of "I" and "Y", the aggregate demand for money equals aggregate supply of money in the economy. (Note that we have the prices as exogenously given and fixed)

R codes

Simple IS-LM framework with simulated data

Equilibrium: The point of intersection of the IS and LM curve is the combination of "I" and "Y" for which both the goods and the money market are in equilibrium.

Effects of fiscal policy (or increase in government expenditure)
An increase in the government spending (fiscal expansion) results in the rightward shift of the IS curve. This happens because the autonomous component of the aggregate demand ("A" in the above derivation) increases. Fiscal policy (with exogenous price level) leads to increase in both output and interest rates.

Effect of fiscal policy

Effect of monetary policy (change in money supply)
An increase in the money supply by the central bank (monetary policy) results in the rightward shift in the LM curve. This happens because of the increase in nominal money supply (MS), and since the prices are exogenous the entire curve shifts to the right. Monetary policy (with exogenous prices) leads to fall in interest rates and rise in the output.

Effect of monetary policy

Mixture of monetary and fiscal policy
Any desired level of output and interest rates can be achieved by using both these policies in tandem, as illustrated below.

Effect of both fiscal and monetary policy

Aggregate demand curve (AD curve)
Now if we introduce prices in the picture we can trace out different combinations of "P"(prices) and "Y" for which both the goods and the money market are in equilibrium. In order to achieve this we have endogenised the computation of prices (P) (by taking ms = (nominal money)/(prices))  This represents an important relationship between aggregate price level and output for the economy. The downward sloping AD curve should not be confused with the demand curve for a good (as in microeconomics). Although both are downward sloping, the reasons for the negative relation between the prices and output demanded are different in the 2 cases. In microeconomics if the price of a good increases less of it is demanded, ceteris paribus. However in the case of AD curve, this negative relation is established by the interplay between the goods and the money market that ensures that markets clear.

This completes the story from the demand side, as to how we arrived at the aggregate demand curve for the economy. Note that we have taken a simplified version of the equations to make our point clear and to make it useful for non-econ students too. We could introduce government taxes to make the equations more realistic (and complicated) but the math looks cleaner this way and anyways the intuition remains the same even after incorporating taxes.

The next thing that we need to do is to arrive at the aggregate supply curve (labor side) for the economy, which we shall take up in the next post. Once we have presented both the ideas of aggregate demand and aggregate supply, we would be in a position to better understand the above discussion about the classical and Keynesian school of thought.

Memoization in R : Illustrative example

I came across a nice problem at project euler that gave me sense of satisfaction that was unusual, I think that because I don't usually get the solutions right the first time as I did in this case. Anyhow, I shall try and decode the R codes that I used in simple English language and Mathematics.



Let me first illustrate the brute force method, that is usually the method used by novice coders like myself. The idea here is to find the largest number below 1 million that gives the maximum number of the above mentioned iterations.



So what I have done above is simply performed the iteration for each and every integer from 1 to 1 million and using a counter variable kept a track of which number gave me the largest number of iterations and recorded the corresponding number, which is what we needed in the end. The idea was straight forward the only challenge was to come up with that cute function (which we now see is not that challenging after all).

Well, now that the novice part is done lets get to what Utkarsh (my pro bro) had to say about this. My codes took ~ 701.75 seconds to run (on my Sony vaio VPCCW12EN), this was completely fine by me. Utkarsh shrugged in his usual nonchalant manner at my codes and came up with an awesome algorithm to optimize the above calculation and saving some precious time (which I think he referred to as Memoization). The idea that he worked on was that since in many cases we would already have computed the number of iterations there was no need to keep computing then again. Suppose in the example in the question we see that 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1. Now in the computation of 13 if say we already have that 10 will further iterate say 6 times we would not have to go all the way to 1. Similarly even for 10 if we know that 5 further iterates 5 times we don't need to go all the way back till 1. This would be more clear when we take a look at the codes.



The above codes, courtesy Utkarsh, took ~ 50 seconds. As it turns out I was 1,390% inefficient as compared to this optimal algorithm. I would glad to know if there is any other optimization technique (apart from using super computers) that might reduce the computational time, please share if you can find a better way of coding this in R. 

Modelling returns using PCA : Evidence from Indian equity market

As my finance term paper, I investigated an interesting question where I tried to identify macroeconomic variables that explain the returns on equities. Much of the debate has already taken place on this topic which has given rise to two competing theories of asset pricing viz. CAPM (capital asset pricing theory or single factor model) and APT (arbitrage pricing theory or multi-factor model). Here is a brief discussion on the two in my previous post. In this post I would like to discuss my approach to answering this question in the context of Indian stock market.

Methodology:

• Companies that have been actively traded on NSE stock exchange for the past 10 years (218 companies) were selected and their daily stock returns data for these 10 years was taken from PROWESS. 
• Using PCA, first 10 components from the returns data of the 218 companies was extracted. More on PCA in my previous post, here. 
• These components were then separately regressed first on NIFTY returns (first regression) 
• Then these components were regressed on NIFTY returns, MIBOR rate changes, and INR/USD exchange rate changes (second regression).
• The explanatory power of the 2 regressions were compared using a F-statistic. (refer to pg. 10 in the paper attached in the end of the post)

Findings and R codes:
We start with calculating the PCA of the returns on the 218 companies daily return data, then employing the 2 regressions, then comparing the 2 regressions using a F-statistic. F-stat tells us if there is any additional explanation offered when we include macroeconomic variables (viz. MIBOR, INR/USD) in our equation.



The results that I obtained pose an interesting observation. We find that the F-stat is significant at 5% for 7 out of the 10 regressions, meaning that out of the 10 regressions (each regression with a separate component) we find statistically significant addition in the explanatory power of the model after adding the macroeconomic variables. Therefore, on statistical ground I can argue that a multi factor model (APT) is preferable over a single factor model (CAPM) for modelling stock returns in the case of Indian equity market. This assertion, if holds true, can have reaching implications for asset pricing for Indian securities. Let me explain why. The principal components (that are the dependent variables in the model) are essentially the common factor across all the companies stock returns with the idiosyncratic effects discounted, so any variables that explains this common component would be the systematic risk (think why!). Now we can relate it to the debate between the CAPM and APT guys. If the CAPM guys were correct, I would obtain no additional explanation in my model after adding the macroeconomic variables i.e their assertion that the market risk (market beta) capture the entire systematic risk holds true.

The results, however, suggest that in 7 out of 10 regressions there is statistically additional explanation offered by the macroeconomic variables. Well, so we can out-rightly reject the applicability (of the much prevalent) CAPM in the case of Indian equities. Or is there something amiss? Now if I closely look at the absolute increase in the explanatory power by looking at the Adjusted-R-squared values before and after the addition of the macroeconomic variables, the absolute increase in all the cases is < 1% (refer to pg. 11 in the paper at the end of this post). Therefore, although we obtain statistical efficiency after the addition of the variables, the economic efficiency (intuition) is called to question. Is it worth while to complicate our model with additional macroeconomic variables, when we can simply have the market rate used as a reasonable proxy for all the variables? And all this just to prove a point that we have macro-variables that can provide 0.5% additional explanation in our model? This takes us back to the eternal debate of statistical vs economic efficiency, what is more important? Is the above result robust enough (on economic intuition) to question the much used, simple and powerful CAPM? Is there a threshold even in statistical efficiency to ensure economic efficiency? These are some questions that still linger on in my mind.

If we view the above result with this caveat of economic efficiency then there is reason for us to believe that a single factor model would be a preferable way to model stock returns. There are, however, evidences in the literature to suggest that multi factor (APT) is a superior way of modelling returns, but the identification of these "multi factors" remains a contentious issue among the researchers. In some desperate attempts to refute CAPM, researcher extracted principal components from a number of macroeconomic variables as the input to the PCA. This resulted in factors that had no economic intuition at all, that were then used as independent variables in explaining the returns. The APT (Arbitrage pricing theory) is a 'theory', whereas CAPM is a 'model' that approximates reality. So even if in reality there are multiple factors that give rise to the returns signals as we see them, the identification of these factors is not a trivial exercise as we have seen above. Statistically we managed to overturn the CAPM in the context of Indian equity markets but in term of economic intuition the results do not seem to be that promising. Therefore, the above exercise tells us exactly why people still stick to the evergreen CAPM as an asset pricing model.

In case you wish to replicate the exercise the data can be obtained from here: Returns_CNX_500, Nifty_returns, MIBOR, Exchange_rates.

Here is the full text of my paper. Feedback are welcome. 

Movement around the mean "Stationary" OR "Unit root"

The idea of modelling the time series of GNP, and other macroeconomic variables, data for US as a trend stationary (TS) process was brought into question by Nelson and Plosser in their groundbreaking research paper in 1982. Their research paper marked a paradigm shift in the way time-series econometrics was done post the 80's. The profound idea that prompted them to look for an alternative to the prevalent TS process, was that the series of GNP does not have any tendency to return back to a time trend following a shock. This means that following a shock (for example technological innovations), the series keeps moving away from the time trend rather than return back to it. If the series keeps moving away from the time trend, movements of the series would not be captured by a trend-stationary model.

This marked a radical change which transformed the idea of stationarity to include another class of processes, difference stationary (DS) processes. More on this in my previous post. But as a student of basic time series the phenomenon of non-stationarity was not very easy for me to digest. Does it mean that if a series fluctuates around a mean, is it necessarily stationary? The answer happens to be No (now that I have completed the course I can proudly and confidently answer that question). According to the definition of stationarity, a series is stationary if any group of consecutive data points in the series, have the same mean. Sounds confusing? Let me illustrate this using the example of 2 Indian macro series and some R codes. The daily 3-month MIBOR rates and the daily INR/USD exchange rates for the past 10 years.

###############################

# Access the relevant files ###

###############################

mibor <- read.csv("MIBOR.csv", na.strings="#N/A")

exchange <- read.csv("Exchange_rates.csv", na.strings="#N/A")

nifty <- read.csv("Nifty_returns.csv")

#################################

## Dealing with missing values ##

#################################

## Dealing with blanks in the MIBOR rates ##

mibor[, 2] <- approx(as.Date(mibor$Dates, '%d-%b-%y'), mibor[ ,2], as.Date(mibor$Dates, '%d-%b-%y'))$y

for(k in 2:nrow(mibor))  # Calculating the %age change

{

  mibor$Change1 <- diff(mibor$MIBOR) / mibor$MIBOR[-length(mibor$MIBOR)]

}

## Dealing with blanks in the exchange rates ##

exchange[, 2] <- approx(as.Date(exchange$Year,'%d-%b-%y'), exchange[ ,2], as.Date(exchange$Year, '%d-%b-%y'))$y

exchange$Change <- as.numeric(exchange$Change)

for(j in 2:nrow(exchange)) # Calculating the %age change

{

exchange$Change <- diff(exchange$Exchange.rates)/exchange$Exchange.rates[-length(exchange$Exchange.rates)

}

## Plotting the variables ##

png("indep_var_ns.png", width = 480, height = 480)

par(mfrow = c(2, 1))

plot(as.Date(mibor$Dates,'%d-%b-%y'), mibor$MIBOR, xlab= "Date", 

     ylab= "3-month MIBOR rates (%age)", type='l', col='red', 

     main="3-month MIBOR rates")

abline(h = 0, lty = 8, col = "gray")

plot(as.Date(exchange$Year, '%d-%b-%y'), exchange$Exchange.rates, xlab= "Date", 

     ylab= "IND/USD Exchange rates", type='l', col='red', 

     main="IND/USD Exchange rate")

abline(h = 0, lty = 8, col = "gray")

dev.off()

Eyeballing the above plots one can see that the series do not have any trend in them, as in the series are moving more of less about a mean. But if we look at the MIBOR for example, the mean of the series is different in the period 2000-02 and different for 2003-04. This is the catch here, which I think is quite probable to be overlooked by many. A unit root would also cause long forays away from the mean, so to conduct a test for non-stationarity we shall check if the above series has a unit root in the auto-regressive (AR) polynomial using the ADF test. And now that we can see that the mean is changing substantially over the time horizon, we would expect there to be a unit root in the series. Let us see what the results have to show.

> adf.test(exchange$Exchange.rates)

Dickey-Fuller = -1.9266, Lag order = 13, p-value = 0.6094 ## Cannot reject the null of non-stationarity

alternative hypothesis: stationary

> adf.test(mibor$MIBOR)

Dickey-Fuller = -2.1925, Lag order = 13, p-value = 0.4968 ## Cannot reject the null of non-stationarity
alternative hypothesis: stationary

> adf.test(nifty$S...P.Cnx.Nifty)

Dickey-Fuller = -11.8633, Lag order = 13, p-value = 0.00 ## Can reject the null of non-stationarity
alternative hypothesis: stationary

So we see that the null of unit root cannot be rejected for MIBOR and INR/USD, but the null is rejected for NIFTY returns. Why its rejected for NIFTY is because the fluctuations around the mean are of a very high frequency, so even if we took 2 different time periods the statistical difference between their means would be negligible. Thus the NIFTY returns gives us a stationary series. MIBOR and INR/USD series are also made stationary by taking first difference of the series. The stationary plot look like:

## Plot for the %age changes of the variables:

png("indep_var.png", width = 480, height = 480)

par(mfrow = c(3, 1))

plot(as.Date(mibor$Dates,'%d-%b-%y'), mibor$Change1, xlab= "Date",

     ylab= "Change in 3-month MIBOR(%age)", type='l', col='royalblue',

     main="%age change in MIBOR rates")

abline(h = 0, lty = 8, col = "gray")

plot(as.Date(nifty$Date,'%d-%b-%y'), nifty$S...P.Cnx.Nifty, xlab= "Date",

     ylab= "NIFTY returns(%age)", type='l', col='royalblue',

     main="NIFTY returns")

abline(h = 0, lty = 8, col = "gray")

plot(as.Date(exchange$Year, '%d-%b-%y'), exchange$Change, xlab= "Date",

     ylab= "IND/USD Exchange rates change(%age)", type='l', col='royalblue',

     main="IND/USD Exchange rate changes(%age)")

abline(h = 0, lty = 8, col = "gray")

dev.off()

So there are 2 takes from the exercise above (1) Series fluctuating about a mean need not necessarily be stationary (empirically shown) (2) 3-month MIBOR and INR/USD exhibit unit roots in the given (10 year daily) sample for India. The first point might be a trivial statement for advanced econometricians, but for the novice and the amateurs I think this would serve as a good basic exercise.

In case you wish to replicate the exercise, data can be obtained from here: MIBOR, INR/USD, NIFTY. 

Create your own Beamer template

For the past couple of days, I had been searching for a tutorial that would show how to create a custom Beamer template. I found some great resources and some really great customized templates (I have listed the ones that I referred to below) but none indicated how should I go about it. There was just lines and lines of code and for someone like me, who is programmatically challenged, the task of creating my own template seemed very daunting. So I slowly started reading some of code that I found and tried to made sense of. I did eventually succeed but not entirely.

I did manage to create my own template but is relatively much simpler as compared to the ones that I saw over the internet. But this did get me started, so I am hoping that anybody who's looking for a decently documented procedure on the topic would find this post helpful. The code is relatively simple, both to understand and execute (I hope).

One of the best documents to understand Beamer is the Beamer user guide. It explains the Beamer mechanisms very well and I used it as one of the major references while preparing this template. According to the guide, Beamer presentations have five flavours of themes

1. Presentation themes - Every aspect of the presentation is detailed here. The colour, the font, the way the bullets look. The way the enumeration goes. The size and position of the logo etc.
2. Colour themes - Just of the colour details. This can be created as a separate file from the presentation theme and then later called with other presentation themes. I will explain this below. In fact, let's just enumerate the other three and jump to the explanation of how all this connects.
3. Font themes
4. Inner themes - Design the elements that are "inside" the frame like the environments, theorems, blocks etc.
5. Outer themes - Design the outer space of the frame like the headline, footline, sidebar etc. 

To create a Beamer theme we need to specifiy four types of details-- colour, font, inner, outer. These details can be mentioned in the presentation theme itself or can be created as different files and then called in the TeX document. It is actually advisable to create them as separate files since this will allow us to use these specific themes with other presentation themes as well by just calling these in the TeX document. Additionally, it is also a more efficient way of working. For example, let's say that we like the overall feel of the Pittsburgh theme but just want to change the colours from blue to black. Then we need to just create the colour theme file and call it in the TeX document. This wouldn't be possible if we had just created one file with all the details. Let's just try to create a simple theme and see how it goes.

%%%%%%%%%%-------------------------------------------%%%%%%%%%%%%

%                             File created: 9 Nov, 2011

%
% This beamer style file was created just for experminetation and learning.
% The file is pretty much self-explanatory and as of 10 Nov, 2011, pretty much bug free (This is because I really don't know what "bugs" are and/or how to find them.
% Howvere, in case you find and bugs, issues, or have any suggestions/comments, please feel free to contact me at programming-r-pro-bro.blogspot.com
%
%
%
%%%%%%%%%%-------------------------------------------%%%%%%%%%%%%


% This style file is a combination of the four files required to create the Moo beamer theme.

% 1. Inner
% 2. Outer
% 3. Color
% 4. Font
% You can directly use this file instead of the using the other theme file "beamerthemeMoo-whole.sty" and calling the 4 styles in the TeX document.




%%% Defining the preamble
\mode<presentation>
\usepackage{pgfcomp-version-0-65}
\usepackage{color}


%%%%%%%%%%%%%%
%%
%% Color theme
%%
%%%%%%%%%%%%%%




%%%%%%
% We need to define a set of colours that will be assigned to various parts of the presentation. I personally believe that Beamer already does have some great themes, so if we really want to build our own, we will have go to the tiniest of details and tweak them.
%%%%%%





%%%%%%
% LaTeX, like R, has a huge set of colours to choose from and there are many ways we can access them.
% Colours in LaTeX are provided by the xcolor pacakge that loads by default. However, the xcolor package by itself does not provide enough breadth in colours.


% More colours can be called by using the "dvipsnames" options while calling beamer in the preamble of the document, e.g., \documentclass[xolor = dvipsnames]{beamer}



% Other than that we can also blend differnt colours to achive the desired colour mix or simply use the RGB codes to call that particular color.


% For blending two colours, we need to specify the two colours (Duh-uh!) and the percentage share of the two colours in the folowing format: <color1>!<percentage of color1>!<color2>
% If percentage of color1 is specified as x%, then percentage of color2 will automatically be taken as (100 - n)%.




% Why don't we try out a few options and see for ourselves.




% Here we are defining only two colours. We are primarily only going to use the first one.

\definecolor{Ftitle}{rgb}{0, 0, 0} % (rgb - 0, 0, 0) is nothing but black

% Here, we are using "rgb" in small case and this notation for decimal values of "rgb" ranging from 0 to 1

% To specify the integer values of "RGB", ranging from 0 t0 255, we need to use the "RGB" in CAPS or UPPERCASE
% Also, the first curly braces include the name that we assign to the colour combination
\definecolor{Descitem}{RGB}{0, 0, 139}

\definecolor{StdTitle}{RGB}{26, 33, 141}
\definecolor{StdBody}{RGB}{213,24,0}

\definecolor{AlTitle}{RGB}{255, 190, 190}
\definecolor{AlBody}{RGB}{213,24,0}

\definecolor{ExTitle}{RGB}{201, 217, 217}
\definecolor{ExBody}{RGB}{213,24,0}




% Another color for the background canvas using the blending option
%\definecolor{BgShade}{red!30!white}
%%% Important note: While trying this, I found out that a colour cannot be defined this way, it can only be set or called for a particular feature using the \setbeamercolor{}{} command. We will use this below to show this.


%%%%%%
% Assign colours to different constitutens of the presentation as per the requriements.
%%%%%%



% This sets the colour of the title of the presentation and titles of all the slides in the presentation to black.
\setbeamercolor{frametitle}{fg = Ftitle}
\setbeamercolor{title}{fg = Ftitle}

% In case you choose to display the Table of Contents, or the Outline slide.
\setbeamercolor{section in toc}{fg = Ftitle}
\setbeamercolor{section in toc shaded}{fg = Ftitle}

% The colour of all the items, subitems and and subsubitems are set to black.
\setbeamercolor{item}{fg = Ftitle}
\setbeamercolor{subitem}{fg = Ftitle}
\setbeamercolor{subsubitem}{fg = Ftitle}

% This sets the color for each item heading of the description environment.
\setbeamercolor{description item}{fg = Descitem}

% NOTE: Setting the color black for all the items also sets in black for other environemnts like enumerate.

% We also need to fix the colours for captions for figures and tables.
\setbeamercolor{caption}{fg = Ftitle}
\setbeamercolor{caption name}{fg = Ftitle}

% In addition, we can also change the background colour of the slides depending on pur requirement.
% \setbeamercolor{background canvas}{bg = blue!5}
% We have commented out this command because this is just for illustrative purposes and has not been used to define the background colour of the slides.




%%%%%%
% Now, there are three types of boxes in beamer:
% 1. Simple, or standard block, which can be invoked using definition or theorem
% 2. Alert block
% 3. Example block
% We will customize all these blocks based on our requirements


% Standard block

\setbeamercolor{block title}{fg = Descitem, bg = StdTitle!15!white}

\setbeamercolor{block body}{bg = StdBody!5!white}

% Alert block

\setbeamercolor{block title alerted}{bg = AlTitle}

\setbeamercolor{block body alerted}{bg = AlBody!5!white}

% Example block

\setbeamercolor{block title example}{bg = ExTitle}

\setbeamercolor{block body example}{bg = ExBody!5!white}

%%%%%%
% And one final thing, the colour of the text
\setbeamercolor{normal text}{fg = Ftitle}






%%%%%%%%%%%%%%
%%
%% Font theme
%%
%%%%%%%%%%%%%%


%%%%%%
% Here we are using default fonts
\usefonttheme{professionalfonts}

% Font for the presentation title
\setbeamerfont{title}{size = \huge}

% Font of the frame titles
\setbeamerfont{frametitle}{size = \Large}


%%%%%%%%%%%%%%
%%
%% Inner theme
%%
%%%%%%%%%%%%%%


%%%%%%
% Here we am using the rounded theme for the overall "feel" of the presentation. You can change the specific details by editing that particular option as we have done below.
\useinnertheme{rounded}

% Instead of rounded circles, we will use triangles as the indicator for items.
\setbeamertemplate{itemize items}[triangle]

% The default option for enumerate environment removes the circles around the numbers provided by the "rounded" inner theme. Just simple numbers remain.
%\setbeamertemplate{enumerate items}[default]


%%%%%%%%%%%%%%
%%
%% Outer theme
%%
%%%%%%%%%%%%%%


%%%%%%
% The outer theme takes the most amount of effort and time to customize.


%%%%%%
% In the outer theme, we will try to do the following:
% 1. Change the headline by putting a logo and a horizontal line
% 2. Change the footline and include custom information depending on our requirements
% 3. Organize the presentation title and the frame titles


%%%%%%
% Let's start with the headline. The approach that we plan to take for the headline and footline is similar.
% We will first define a new command and then include the command in the \setbeamertemplate{} option.
% I tried a couple of approaches that would make the method more simple but could not come up with one. % In case you do find a more aesthetic approach, please do send it accross. The contact information is on the top of the page.
% First the horizontal line on the top portion of the slides
% Add a horizontal line that runs from left of the slide to the right, just below the logo.
\newcommand{\LogoLine}{%
\raisebox{-12mm}[0pt][0pt]{%
\begin{pgfpicture}{0mm}{0mm}{0mm}{0mm}
\pgfsetlinewidth{0.28mm}
\color{gray}
\pgfline{\pgfpoint{-3mm}{1mm}}{\pgfpoint{10.8cm}{1mm}}
\end{pgfpicture}}}


% Include the line that we just created in the headline
\setbeamertemplate{headline}[text line]{\LogoLine}


% Now the logo. As it turns out. I could not include the logo in the headline. If I tried to, the headline kept shifting downwards.
% Acting a little smart and lazy, I just included the logo in the right sidebar and shifted it up.
\setbeamertemplate{sidebar canvas right}{
\vspace*{3pt}\hspace*{-25pt}%
{\includegraphics[height=28pt]{moo.png}}}





%%%%%
% Now that we have changed the headline, we will need to orient the frame titles in a way that the come at the right stop just above the horizontal line.
\setbeamertemplate{frametitle}{
\vspace*{4mm}\hspace*{-2mm}\insertframetitle}


%%%%%
% Like mentioned above, we will take a similar approach to customize the footline as well and inclued FAA in it.

\newcommand{\Ffootline}{%

\insertsection % The left end of the footline

\hfill

\textit{Moo} % The center

\hfill

\insertframenumber/\inserttotalframenumber} % And the right end

\setbeamertemplate{footline}{%

\usebeamerfont{structure}

\begin{beamercolorbox}[wd=\paperwidth,ht=2.25ex,dp=1ex]{title in head/foot}%

\Tiny\hspace*{4mm} \Ffootline \hspace{4mm}

\end{beamercolorbox}}

%%%%%%

% We will also remove the navigation symbols, which I personally don't find very useful
\setbeamertemplate{navigation symbols}{}
%%%%%%
% Now the toughest part--at least for me--customizing the title page.
% Putting a logo on the title page and text beside it was quite a difficult task and to be fairly honest, my code is not effecient at all. At the very least, it is plain clumsy.
% Though it does solve the purpose (I hate to use this phrase) it is neither "neat" nor "cool".
% Anyway, let's see how we went about it.
% For the title page, we needed a logo on the left, a vertical separater line, and finally a place for the title, author, date etc.



% First, let's create the line
\newcommand{\TitleLine}{%
\raisebox{-12mm}[0pt][0pt]{%
\begin{pgfpicture}{0mm}{0mm}{0mm}{0mm}
\pgfsetlinewidth{0.10mm}
\color{gray}
\pgfline{\pgfpoint{55mm}{0mm}}{\pgfpoint{55mm}{50mm}}
\end{pgfpicture}}}



% Now let's create commands for the title etc., that we can call later

% Title
\newcommand{\MyTitle}{%
\hspace*{60mm}\vspace{-25mm}
\centering \inserttitle}

% Subtitle
\newcommand{\MySubTitle}{%
\hspace*{60mm}\vspace{-25mm}
\centering \footnotesize \textit{\insertsubtitle}}

% Author
\newcommand{\MyAuthor}{
\hspace*{60mm}\vspace{-25mm}
\centering \insertauthor}

% Institute
\newcommand{\MyInstitute}{
\hspace*{60mm}\vspace{-25mm}
\centering \footnotesize \textit{\insertinstitute}}

% Date
\newcommand{\MyDate}{
\hspace*{60mm}\vspace{-25mm}
\centering \insertdate}



% We declare the image that will be used as the logo
\pgfdeclareimage[width = 0.20\paperwidth]{big}{moo.png}



% This is quite a complicated command. We basically create a "beamercolorbox" for each field and invoke the commands that we had created earlier.

\setbeamertemplate{title page}{\TitleLine

\hspace*{11mm}\vspace*{-60mm}

\begin{beamercolorbox}[wd=0.5\paperwidth,ht=0.13\paperwidth]{Title}

\pgfuseimage{big}

\end{beamercolorbox}

%

\begin{beamercolorbox}[wd=\paperwidth,ht=0.06\paperwidth]{Title}

\usebeamerfont{Title}%

\MyTitle

\end{beamercolorbox}

%

\begin{beamercolorbox}[wd=\paperwidth,ht=0.03\paperwidth]{Title}

\usebeamerfont{Title}%

\MySubTitle

\end{beamercolorbox}

%

\begin{beamercolorbox}[wd=\paperwidth,ht=0.06\paperwidth]{Title}

\usebeamerfont{Title}%

\MyAuthor

\end{beamercolorbox}

%

\begin{beamercolorbox}[wd=\paperwidth,ht=0.03\paperwidth]{Title}

\usebeamerfont{head/foot}%

\MyInstitute

\end{beamercolorbox}

%

\begin{beamercolorbox}[wd=\paperwidth,ht=0.07\paperwidth]{Title}

\usebeamerfont{Title}%

\MyDate

\end{beamercolorbox}}

\mode

<all>

Well, this pretty much completes the creation of Moo. I will upload the Sty file, TeX file and sample pdf for direct use and references. 

Also, in case you are trying to create a more complicated theme you should definitely check out some of the resources mentioned below.

1. Most important - The Beamer user guide
2. A great repository of themes here
3. A great introduction to Beamer here
4. Some information that helped
5. Another custom theme here
6. A very good Beamer example

Hope this information helps.

The uploaded file

Style file

TeX File

PDF output