Economics and Statistics

  Let X and Y be column vectors and the sample has the size n: The vector beta contains both the coefficient of X as well as the coefficient for the intercept. This is equivalent to writing this equation: We will assume that the expected value of the error term is 0 (this can be done by construction), and that X is uncorrelated to epsilon. Assume the following is for population data. Let's prove that statement: Since we know that the covariance between X and epsilon is 0, it follows that the expected value of X times the error term must also be 0. Since we do not have access to the actual expected value of the distribution, let's use the sample data instead: The hat on the covariance signifies that this is a MM estimator. Let's use the fact that the sample mean of epsilon is also equal to 0, then:   Using the estimation for the covariance and that the expected value of the error term equals 0: Then we have: To get this result, you must use the properties of the summation op...

Hello I'm finally back from my extended break. I thought that we should start studying Econometrics.  Let's begin by analyzing a simple bivariate regression. Assume that this equation describes the relationship between two variables X and Y. We say that Y is the dependent variable, whereas X is the independent variable. In other words, we assume that Y (the output) depends on X (the input). Epsilon is the error term, it represents other factors that affect Y. The error term must be uncorrelated with the variable X so that we do not need to include them in our regression, and thus the coefficient of X (beta) should not change, even though Epsilon also determines Y.  beta-0 is the constant term. It tells us what would be Y if X=0. beta-1 is the effect on Y if X changes by one unit. To see this, assume X=education is a continuous function, let's take the derivative of Y=wage with respect to X: Thus, if education goes up by one unit, we should expect, on average, wage to go up ...

Hello all, it's been quite some time since I have made any major announcements since the creation of this blog about a year and a half ago. I am currently in graduate school and will soon take a portion of my comprehensive examination (i.e exams in Micro, Macro, and Metrics that will allow me to continue my studies in my graduate program), wish me luck! Thus, I will not be able to post consistently until at least mid-June. The roadmap is as such, if I pass, I will start introductory lessons in Econometrics and Statistics (if not, then I'll have to study until I can retake the comps in August). I hope that once I am done, I will be able to add more advance materials to the blog, such as general equilibrium, indirect utility functions, and game theory/mechanism design for Micro. The Solow, Ramsey, RBC, New Keynesian models, permanent income hypothesis (PIH) and more for Macro. By the way I think I still need to add notes on the IS-LM curves, so I will do that before jumping to t...

Assume that the U.S government wants to start fixing its currency against the Mexican peso.   Government officials estimate that the optimal exchange rate should be at one peso for $0.35.  You are given the following information about the U.S market for Mexican peso: Find how much peso does the U.S needs to buy or sell in order to achieve its targeted exchange rate. Find the original equilibrium exchange rate: let the demand equal to the supply curve. Just like in the previous exercise, we see that one peso is worth half a U.S dollar and there are 6,250,000 pesos demanded at this price. Now, in order to change the exchange rate, realize that the U.S government has access to U.S dollars (for the sake of the argument, assume the central bank is no longer independent). Then, it can only shift the demand for Mexican pesos in the U.S foreign exchange market. Thus, there must only be a movement along the supply curve, set $=0.35 in the supply schedule and solve for p. Now, calculate...

Floating exchange rate:  the equilibrium exchange rate and quantity determined by the foreign exchange market. Anything that shifts the demand or supply curve will change the equilibrium exchange rate. Determinants of demand and supply for a currency: Change in taste and preferences: say Chinese consumers start to prefer American made cars. We should expect the demand for US dollars to go up (in the market for dollars in China) and the supply for the Chinese Yuan to go up as well (in the market for Yuan in the U.S). This will lead to an appreciation of the U.S dollars and a depreciation of the Chinese Yuan. Relative income level: consumers will be more likely to consume goods from countries with lower inflation rates. Relative interest rate: the higher the interest rate (relative to another country), the higher the demand for a country's currency, as investors are interested in this opportunity.  Speculation: expectation about a country's exchange rate among investors. Fo...

  You are given the following demand and supply schedules for the U.S foreign exchange market for Mexican pesos: Identify the equilibrium exchange rate and quantity, and draw the demand and supply curves. Then, what would happen to the value of the pesos if the demand for American goods increases? What should we expect to happen to the U.S current account? Assuming the trade balance is by far the biggest variable and the U.S 's current account was equal to 0 before this sudden increase. The equilibrium must be the point that is shared by both the demand and supply schedules. Thus, one pesos is worth about 0.5 U.S dollars (the exchange rate) and there are 6,250,000 pesos supplied at this point, assuming that one p is worth 10 million pesos. Let's graph the demand and supply curves: If the consumers in Mexico want to consume more American goods, then we should expect the supply curve for pesos to shift to the right. This will cause an appreciation of the dollar and depreciation o...

A currency exchange rate tells us about the value of a currency relative to another currency.  We say that a currency appreciates when its relative value goes up. We say that a currency depreciates when its relative value decreases. In a market that relates two currencies, if one appreciates, it must be the case that the other currency depreciates.  Demand in the foreign exchange market: it represents the quantity of a currency demanded by agents who are holding other currencies. The agents want to buy goods, services, or financial assets from a country whose currency is demanded. The demand must be downward slopping. The weaker the currency, the more attractive the goods produced in that country are, and foreign consumers need to hold more of that currency in order to buy the products. Thus, a change in the exchange rate leads to a movement along the demand curve. Supply in the foreign exchange market: it represents the willingness of people in the country supply to foreigner...

Current account deficit:  this will cause a currency depreciation, as the demand for imported goods is bigger than the foreign demand for exported goods.   This will make it easier for domestic producer to exports their goods as they, the goods, become relatively cheaper on the foreign market. A deficit in the balance of trade implies that the financial account must be positive (remember Xn= S-I). In order to pay for the imported goods, it must be the case that foreign entities own more domestic financial assets. Furthermore, foreign countries will buy domestic government debt. The central bank can try to offset inflation by raising the interest rate which can then attract foreign capital (i.e money). However, in the short term, this will be a monetary contractionary policy. Current account surplus: The domestic currency will appreciate, since foreign demand for exported goods is bigger than the domestic demand for imported goods. Note that domestic savings are then ...

Balance of payment is the sum of three separate accounts and must always be equal to 0 (given no statistical error): Current account: the net flow of funds exchanged for goods and services, and monetary gifts  that flow in and out of a country. This is often used as an indicator for net export. Financial account: also known as the capital account, is the net flow of funds for investment in real assets (direct investments) or financial assets (stocks or bonds) into a nation. Official reserves: in order to balance the two accounts above, a country must have a reserve of foreign money. It measures the net effect of all money flows from the other accounts. If the current account is positive, it must be the case that the sum of the financial account and official reserves is negative. Current account (CA) components: Balance of trade in goods: spending by consumers and firms on imported and exported goods. Exported goods have a positive effect on the current account, whereas importe...

  Economic growth: measures the change in productive capacity of an economy between one period of time and another. If growth is negative, then the economy is experiencing a recession. If growth is positive, then the economy is expending. If the rate of population growth is smaller than the rate of economic growth, then on average people are becoming richer. Measuring economic growth (in discrete terms): Short term vs long term economic growth: we saw that an increase in the aggregate demand curve increases a nation's output only temporally. However, if the aggregate supply changes due to new technology or to a change in the factors of production (excluding wages) then the LRAS would shift, leading to a sustainable long term economic growth.  Note that when the aggregate demand hits the LRAS there is no long term economic growth assuming constant technology and capital levels.  Sources of long term economic growth: Human capital: anything that makes labor more p...

Using our previous exercise, assume that the U.S government is planning on increasing its budget in order to update public infractures, shows this effect on the short term and long term Philips curve. Short term: since we know that the variable G will go up, and we can also safely assume that the C will also increase, the aggregate demand curve must shift to the right. This implies that there will be inflation accompanied with lower unemployment level. Long term: new public infractures implies cheaper transportation cost and more human capital. On the short term model, we can see that the Phillips curve shifts to the left, since the SRAS will shift to the right.   This increase in human capital (via new schools) will also shift the long term Philips curve to the left, implying that the natural rate of unemployment is now lower.

Short run Phillips curve: it represents a trade off between the level of inflation and the level of unemployment in the economy. Any change in the AD curve will produce a negative relationship between the inflation rate and unemployment rate. For example, an expansionary demand side policy will decrease the level of unemployment but increase the inflation rate (in the short run) and vice versa.  The inverse relationship between inflation and unemployment implies that the Phillips curve is downward slopping. It also shows that deflation is related to high unemployment, think about the AD curve shifting to the left.  A shift in AD is just a movement along the Phillips curve. When the SRAS shifts, the Phillips curve shifts in the opposite direction. Long run Phillips curve: the curve is inelastic, implying that there is no trade off between inflation and employment. Shift in SRAS to the right, the AD curve will shift to the left to reach, once again, LRAS. The impact on the short...

  You are in charge of the government budget for the year 2021. You are told that schools and public roads need to be updated, and you estimate an appropriate budget in order to carry out the task. Describe what will happen in the short run to the economy, and what type of inflation do we see? Describe the supply side effect from this policy, explain what happens in the long run. Assume that there is no crowding out effect. First, note that updating public infractures, assuming taxes stay the same, will require an increase in government spending, which will also have an impact on the investment and consumption level in the economy (remember the spending multiplier?). Therefore, the aggregate demand curve will shift to the right. This will cause inflation (i.e positive change in the price level) and a higher output in the short-run.  Note that if we were to assume a large crowding out effect, the short run aggregate supply curve would shift to the left (increase in production c...

  Types of inflation: Demand pull inflation: caused by a rightward shift in the aggregate demand. The increased in consumption among consumers for a limited amount of goods forces prices to rise. When the economy is below full employment, an increase in AD does not drastically change the price level, since the economy is not fully using its resources (workers and machines) and can therefore easily increase production. When the economy is at full employment, an increase in AD causes a large increase in inflation, it becomes relatively more costly to increase production. Methods for reducing demand pull inflation: Contractionary fiscal policies: raising taxes or reducing the government spending, will put downward pressure on the aggregate demand, and thereby reduces its rightward shift. Contractionary monetary policies: Usually, the central bank is the entity that deals with inflation. It does so by increasing the federal funds rate through open market transaction, which decreas...

  As a reminder, in the long run, wages and prices are considered variables in the economy. Thus in periods of economic expansions, workers will demand higher wages due to the pressure on the labor market (higher demand), increasing costs to firms, shifting the SRAS to the left. They opposite effect happens during periods of economic contraction. Expansionary policies have no direct effect on the long run level of full employment. It can be the case that a positive demand side policy also has a positive supply side effect.  Expansionary monetary policy, and its effect on the LRAS: lower interest rates implies more investment which implies more capital (i.e machines), making the workers more productive, leading to economic growth (i.e increasing the level of output in the long run). More investment does shift the AD to the right but so do the LRAS and SRAS curves. Contractionary demand side policies can have an impact on the long run level of full employment if and on...

  Fiscal Policy: government spending and taxation, aimed at expending or contracting the level of macroeconomic activity in a nation. A tax increase (decrease) will raise (lower) households' disposable income, leading to more (less) consumption. Furthermore, firms will increase (decrease) the number of investment as they get to keep a larger (smaller) share of their profits. An increase (decrease) in government spending affects the variable G that defines the aggregate demand. Government spending also leads to a change in household income, affecting the level of consumption. Tax multiplier: just like any multiplier so far, it is just the sum of a converging geometric series: Say the MPC is equal to 70%, then it must be the case that the MPS is 30%. Thus, the tax multiplier, t, is equal to about -2.32. In other words, for every dollar that goes to tax revenue, total spending decreases by about 2.32 dollars. Note that a tax decrease would be negative, and thus have a positi...

  Assume that you have the following information about the federal funds market: Note that the discount rate must be above the federal funds rate in order for the latter to be effective. In other words, the supply curve becomes perfectly elastic at the DR.  Another important point is that the demand for federal funds also becomes perfectly elastic when the interest rate is at or below zero percent. At this rate, commercial banks demand federal funds as much as they can. You are tasked with conducting the optimal open market operation given that the economy just started to experience a downturn. Given available data you conclude that the optimal rate must be at 0.5%.  Solve for the new supply curve, find how many government bonds (in terms of dollars) should the Fed buy or sell. Then, explain what will happen to the aggregate demand on the AD/AS model. solve for the original rho (FFR): Set the FFR equal to 0.5%, and solve for the dollar amount: Assuming that in o...