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计量经济学 Econometrics ECON20110T

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计量经济学 Econometrics ECON20110T

Consider a generic diffusive process governed by
$$
d x=\mu(x) d t+\sigma(x) d w
$$
for some vector-valued process $x$. We typically have a set of observations denoted by
$$
x\left(t_{0}\right), x\left(t_{1}\right), \cdots, x\left(t_{N}\right)
$$
for some set of $N$ times, usually equally spaced, e.g., $t_{i}=i h$ for some time step $h$. Following Yu (2014), we distinguish the following two limiting cases:
$$
\begin{aligned}
&N \rightarrow \infty, h \text { fixed } \
&h \rightarrow 0, N \text { fixed }
\end{aligned}
$$

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ECON20110T COURSE NOTES ：

for $i=1, \ldots, N$ and (instantaneous) covariance structure given by $d w_{i} d w_{j}=$ $\rho_{i j} \sigma_{i} \sigma_{j} d t \equiv X_{i j} d t$. Consider now the sample covariance of $T+1$ discretely observed realizations:
$$
\bar{v}=\frac{1}{T} \sum_{i=1}^{T} \Delta z_{i} \Delta z_{i}^{T}-\bar{\mu} \bar{\mu}^{T}
$$
where $\Delta z_{i} \equiv z_{i}-z_{i-1}$ and the sample mean $\bar{\mu}$ is given simply by
$$
\bar{\mu}=\frac{1}{T} \sum_{i=1}^{T} \Delta z_{i}=\frac{1}{T}\left(z_{T}-z_{0}\right)^{71}
$$








高级统计 Advanced Statistics ECON20072T

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高级统计 Advanced Statistics ECON20072T

$$
f\left(x_{0}\right)-f\left(\lambda x_{0}+(1-\lambda) x_{1}\right) \leqq(1-\lambda) \cdot \nabla f\left(\lambda x_{0}+(1-\lambda) x_{1}\right) \cdot\left(x_{0}-x_{1}\right)
$$
and
$$
f\left(x_{1}\right)-f\left(\lambda x_{0}+(1-\lambda) x_{1}\right) \leqq-\lambda \cdot \nabla f\left(\lambda x_{0}+(1-\lambda) x_{1}\right) \cdot\left(x_{0}-x_{1}\right) .
$$
Therefore
$$
\lambda f\left(x_{0}\right)+(1-\lambda) f\left(x_{1}\right) \leqq f\left(\lambda x_{0}+(1-\lambda) x_{1}\right) \text { for every } \lambda \in[0,1] .
$$

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ECON20072T COURSE NOTES ：

Hence, from the assumed quasi-concavity of $g_{\AA}(x)$,
$$
g_{j}\left(\lambda x_{1}+(1-\lambda) x_{2}\right) \geqq 0 \quad(j=1, \ldots, m) \quad \text { for any } \lambda \in[0,1] .
$$
which shows the convexity of $F$. Since $F$ is convex and $x_{0}$ is optimal for (I), for any $x \in F$ and every $\lambda \in[0,1]$ we have
$$
f\left(x_{0}\right) \geqq f\left(\lambda x+(1-\lambda) x_{0}\right)=f\left(x_{0}+\lambda\left(x-x_{0}\right)\right) .
$$
In view of the differentiability of $f$ and the optimality of $x_{0}$, we have further $\nabla f\left(x_{0}\right) \cdot\left(x-x_{0}\right)+\alpha\left(x_{0}, \lambda\left(x-x_{0}\right)\right) \cdot\left|x-x_{0}\right| \leqq 0 \quad$ for any $\lambda \in(0,1] .$








宏观经济学 Macroeconomics ECON10252T/ECON10262T/ECON20022T/ECON20032T/ECON20262T/ECON20532T/ECON30032T/ECON30532T

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这是一份manchester曼切斯特大学ECON10252T/ECON10262T/ECON20022T/ECON20032T/ECON20262T/ECON20532T/ECON30032T/ECON30532T作业代写的成功案例

宏观经济学 Macroeconomics ECON10252T/ECON10262T/ECON20022T/ECON20032T

$$
\pi_{u t}=u_{u t} Y_{u t}-g r_{i t} K_{u}=\left(u_{i t} \phi-g r_{u t}\right) K_{u t}
$$
and expected profit is $E\left(\pi_{i t}\right)=\left(\phi-g r_{i t}\right) K_{i t}$.
In this economy, firms may go bankrupt as soon as their net worth becomes negative, that is $A_{i t}<0$. The law of motion of $A_{i t}$ is:
$$
A_{u}=A_{u t-1}+\pi_{u \prime}
$$
that is, net worth in previous period plus (minus) profits (losses). Making use of (4.16) and (4.17), it follows that the bankruptcy state occurs whenever:
$$
u_{i t}<\frac{1}{\phi}\left(g r_{i t}-\frac{A_{t-1}}{K_{i t}}\right) \equiv \bar{u}_{i t} .
$$

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ECON10252T/ECON10262T/ECON20022T/ECON20032T/ECON20262T/ECON20532T/ECON30032T/ECON30532T COURSE NOTES ：

$$
\frac{A_{i j}}{x_{i} x_{j}}=\frac{\tau_{i}}{x_{i}} \text { for any } i, j \in J_{x^{*}}
$$
Similarly,
$$
\frac{A_{r s}}{X_{r} \cdot X_{s}}=\frac{t_{s}}{X_{s}} \text { for any } r, s \in J_{\boldsymbol{x}} .
$$
Since the indices $i$ and $s$ are in $J_{x}$,
$$
A_{\text {is }}=t_{s} x_{i}=\tau_{i} x_{s} \text {, }
$$
which in turn implies that
$$
\frac{\tau_{i}}{x_{i}}=\frac{t_{s}}{x_{s}} .
$$








经济数学入门 Introduction to Mathematical Economics ECON10192T

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宏观经济分析 Macroeconomic Analysis 2 ECON10182T

Conversely, if $\boldsymbol{A x} \leqq \mathbf{0}$ then $-\boldsymbol{A} \boldsymbol{x} \in L \cap \mathbb{R}{+}^{m}$. Hence there exists $\xi \geqq \mathbf{0}$ such that $-\boldsymbol{A x}=\boldsymbol{Y} \xi$. Therefore $\boldsymbol{A x}=-\sum{i=1}^{r} \xi_{i} \boldsymbol{y}{i}=\sum{i=1}^{r} \xi_{i}\left(A \boldsymbol{x}{i}\right)$ so that $A\left(x-\sum{i=1}^{r} \xi_{i} x_{i}\right)=0$. It follows that $\left(x-\sum_{i=1}^{r} \xi_{i} x_{i}\right) \in \hat{X}$ and
$$
\boldsymbol{x}-\sum_{i=1}^{r} \xi_{i} \boldsymbol{x}{i}=\sum{i=1}^{s} \mu_{i} \hat{x}{i} \quad \text { for some } \mu{i} \geqq 0 .
$$
Thus
$$
\begin{aligned}
\boldsymbol{x} &=\sum_{i=1}^{r} \xi_{i} \boldsymbol{x}{i}+\sum{i=1}^{3} \mu_{i} \hat{\boldsymbol{x}}_{i} \
&=\boldsymbol{B}\left(\begin{array}{l}
\xi \
\boldsymbol{\mu}
\end{array}\right) \in K(\boldsymbol{B})
\end{aligned}
$$

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ECON10192T COURSE NOTES ：

$$
\frac{A_{i j}}{x_{i} x_{j}}=\frac{\tau_{i}}{x_{i}} \text { for any } i, j \in J_{x^{*}}
$$
Similarly,
$$
\frac{A_{r s}}{X_{r} \cdot X_{s}}=\frac{t_{s}}{X_{s}} \text { for any } r, s \in J_{\boldsymbol{x}} .
$$
Since the indices $i$ and $s$ are in $J_{x}$,
$$
A_{\text {is }}=t_{s} x_{i}=\tau_{i} x_{s} \text {, }
$$
which in turn implies that
$$
\frac{\tau_{i}}{x_{i}}=\frac{t_{s}}{x_{s}} .
$$








宏观经济分析 Macroeconomic Analysis 2 ECON10182T

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宏观经济分析 Macroeconomic Analysis 2 ECON10182T

where $S_{M S K}=\left[1-\gamma \rho \alpha^{\beta}\right]^{-1}$ is the corresponding survivor function, the location parameter $\gamma$ is real, and the sample space for $x$ is $(0, \infty)$ for $\gamma<0$ and $\left(0,(\alpha \gamma)^{-(\beta-1)^{-1}}\right)$ for $\gamma>0$. Besides correcting for unobserved heterogeneity, the additional parameter $\gamma$ allows the hazard function to be nonlinearly monotonic increasing $(\beta>1, \gamma>0)$, nonlinearly monotonic decreasing $(\beta<1, \gamma<0)$, bathtub shaped $(\beta<1, \gamma>0)$, unimodal $(\beta>1, \gamma<0)$ or constant $(\beta=1, \gamma=0$ ). Finally, when $\beta>0$ and $\gamma \leq 0$ the generalized Weibull corresponds to the Burr type XII distribution.

For both models parameters’ estimation has been conducted by means of Maximum Likelihood. The log-likelihood function for a series of expansions (contractions) with observed magnitude $x_{i}$ is:

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ECON10182T COURSE NOTES ：

Let the random variable $S$ to follow a Pareto distribution with parameter $\alpha$. Thus, the probability distribution of $\log (S)$ is:
$$
\operatorname{Pr}(\log (S) \geq k)=\operatorname{Pr}(S \geq \exp (k)) \propto(\exp (k))^{-\alpha}=\exp (-\alpha \mathrm{k})
$$
that is, an exponential distribution with parameter $\alpha$. In other terms, $\log (S)$ follows an exponential distribution with probability density function equal to:
$$
E\left(\log (S), \alpha^{-1}\right)=\frac{1}{\alpha} \exp \left(-\frac{\log (S)}{\alpha}\right)
$$
In the case of independent exponential variables, it is simple to prove that a Laplace distribution regarding growth rates emerges by making use of the convolution theorem and its relation with the characteristic function. In fact, the characteristic function of two independent exponential distributions $z_{j}, j=1,2$, with parameter $\alpha^{-1}$ is:
$$
C_{z j}(\gamma)=(1-i \alpha \gamma)^{-1}
$$








微观经济理论原理 Principles of Microeconomic Theory ECON10172T

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微观经济理论原理 Principles of Microeconomic Theory ECON10172T

If we assume $V=2$ is the initial utility level, this loss (because $P_{Y}=1$ ) is given by
$$
\text { loss }=4(1)^{5}-4(.25)^{5}=2,
$$
which is exactly what we found in Example 5.3-when $P_{X}$ rises to 1, expenditures must rise from 2 to 4 to keep this person from being made worse off. If the utility level experienced after the price rise is believed to be the more appropriate utility target for measuring the welfare loss, then $V=1$ (see Example 5.3) and the loss would be given by
$$
\text { loss }=2(1)^{5}-2(.25)^{5}=1 .
$$
If the loss were evaluated using the uncompensated (Marshallian) demand function
$$
X=d_{X}\left(P_{X}, P_{Y}, I\right)=\frac{I}{2 P_{X}}
$$
the computation would be
$$
\begin{aligned}
\text { loss } &=\int_{.25}^{1} \frac{I}{2 P_{X}} d P_{X} \
&=\left.I \frac{\ln P_{X}}{2}\right|_{.25} ^{1}=0-(-1.39)=1.39,
\end{aligned}
$$

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ECON10172T COURSE NOTES ：

Moving the terms in $\lambda$ to the right and dividing the first equation by the second yields
$$
\begin{aligned}
\frac{1}{X} &=\frac{P_{X}}{P_{Y}} \
P_{X} X &=P_{Y} .
\end{aligned}
$$
Substitution into the budget constraint now permits us to solve for the Marshallian demand function for $Y$ :
$$
I=P_{X} X+P_{Y} Y=P_{Y}+P_{Y} Y
$$
Hence,
$$
P_{Y} Y=I-P_{Y} .
$$
This equation shows that an increase in $P_{Y}$ must decrease spending on good $Y$ (that is, $\left.P_{Y} Y\right)$. Therefore, since $P_{X}$ and $I$ are unchanged, spending on $X$ must rise. So
$$
\frac{\partial X}{\partial P_{Y}}>0,
$$
and we would term $X$ and $Y$ gross substitutes. On the other hand, Equation $6.18$ shows that spending on $Y$ is independent of $P_{X}$. Consequently,
$$
\frac{\partial Y}{\partial P_{X}}=0
$$








震动与波浪 Vibrations & Waves PHYS10302T

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震动与波浪 Vibrations & Waves PHYS10302T

The speed of the rotating mass is the dramference of the circle divided by the period, $v=2 \pi A / T$, its acceleration (which is directly inward) is $a=v^{2} / r$, and Newton’s second law gives $a=F / m=(k A+F) / m$. We write $f_{\text {ra }}$ for $(1 / 2 \pi) \sqrt{k / m}$. Straightforward algebra yields
$$
\frac{F_{r}}{F_{s}}=\frac{2 \pi m}{b f}\left(f^{2}-f_{r s}^{2}\right)
$$
This is the ratio of the wasted force to the useful force, and we see that it becomes zero when the system is driven at resonance.
The amplitude of the vibrations can be found by attacking the equation $|F|=b v=2 \pi b A f$, which gives
$$
A=\frac{\left|F_{\mathrm{t}}\right|}{2 \pi b f} .
$$
However, we wish to know the amplitude in terms of $|F|$, not $|F|$. From now on, let’s drop the cumbersome magnitude symbols. With the Pythagorean theorem, it is easily proven that
$$
F_{s}=\frac{F}{\sqrt{1+\left(\frac{F_{r}}{F_{s}}\right)^{2}}},
$$

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PHYS10302T COURSE NOTES ：

$$
\begin{aligned}
a &=F m \
&=4 T h / \mu w^{2} .
\end{aligned}
$$
The time required to move a distance $b$ under constant acceleration $a$ is found by solving $h=\frac{1}{2} a t^{2}$ to yield
$$
\begin{aligned}
t &=\sqrt{2 h / a} \
&=w \sqrt{\frac{\mu}{2 T}} .
\end{aligned}
$$
Our final result for the velocity of the pulses is
$$
\begin{aligned}
|v| &=w / t \
&=\sqrt{\frac{2 T}{\mu}} .
\end{aligned}
$$








中级管理会计 Intermediate Management Accounting BMAN21040

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中级管理会计 Intermediate Management Accounting BMAN21040

If in the previous example the rates were compounded semiannually, then the annuity would be $\$ 2,500\left(5,000 \times \frac{1}{2}\right)$ instead of $\$ 5,000$, the periods would be 8 instead of 4 , the market rate would be $6 \%$ instead of $12 \%$, and the calculation would be as follows:$\times \$ 2,500$
The bond selling price is:
$\frac{15,524}{\$ 46,895}$

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BMAN21040 COURSE NOTES ：

A $\$ 100,000$ non-interest-bearing note is issued for a machine whose fair market value is $\$ 60,000$. Since it is unlikely for anyone to pay $\$ 100,000$ for an item worth only $\$ 60,000$, we say that the extra $\$ 40,000$ is really “hidden interest.” Recording the machine at $\$ 100,000$ would be overstating the asset Machine and understating Interest Expense. The correct entry should be:
$\begin{array}{lcc}\text { Machine } & 60,000 & \ \text { Discount on Note } & 40,000 & \ \text { Notes Payable } & & 100,000\end{array}$








财务报告和责任制 Financial Reporting and Accountability BMAN21020

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财务报告和责任制 Financial Reporting and Accountability BMAN21020

Issue of new shares. In this case the increases of share capital and share premium reserve are balanced by an increase in cash and debtors
Revaluation of properties. In this case the increase of the revaluation reserve is balanced by an increase in value of the non current assets
Changes in value of certain financial instruments. This occurs when some financial instruments, such as shares, derivatives, etc., are treated at the ir ‘fair value through profit and loss’. The matter is regulated by IAS 39 and is, at the moment of writing this study guide, subject to a ‘relaxation’ of this rule
Changes due to translation from foreign currencies. This derives from the changes in value of assets and liabilities which are denominated in different currencies in the subsidiaries that are part of the reporting entity and had to be ‘translated’ into the reporting currency. This translation might have created losses of gains from one year to the other.

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BMAN21020 COURSE NOTES ：

The formulae are as follows:
Current ratio $=$ current assets / current liabilities
Quick ratio $=$ (current assets -inventory) $/$ current liabilities
Debtors’ days = (trade debtors / sales) X 365
Creditors’ days $=$ (trade creditors / purchases) X 365
Inventory’s days $=$ (inventory / cost of sales) $\mathrm{X} 365$
These ratios are not free from inconsistencies and limitations in their use. Among others: (i) they refer to trade creditors and debtors, whilst we are interested in the whole of the cash flows, but this makes the results more reliable and meaningful; (ii) purchases are normally not given in the account, hence they must be constructed starting from cost of sales and adjusting for amortisation, depreciation and variation of inventories (see cost of sales in chapter 4 on income statement); and (iii) they should refer to more representative values of the debtors, creditors and inventory than the closing ones, e.g. annual averages.








金融市场和机构 Financial Markets and Institutions BMAN21011

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金融市场和机构 Financial Markets and Institutions BMAN21011

Thus it follows that the value of the whole stream of payments is the sum of this progression. If $P$ is the present value or price of the bond, then
$$
P=\sum_{t=1}^{n} C \times \frac{1}{(1+i)^{t}}
$$
In the case of an irredeemable bond, the payments go on for ever and $t$ tends to infinity. This means that the series
$C \times \frac{1}{(1+i)^{t}}$
is converging on zero and the present value $P$ of the sum of the series can be more conveniently written as
$$
P=C / i
$$
This can be confirmed by taking the coupon of any undated government bond from the Financial Times and dividing it by the current long-term rate of interest.

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BMAN21011 COURSE NOTES ：

However, most bonds in fact mature and so our formula has to include a valuation of the payment received on maturity. In this case $P$ is found as follows:
$$
P=\sum_{t=1}^{n} C \times \frac{1}{(1+i)^{t}}+\left[M \times \frac{1}{(1+i)^{n}}\right]
$$
where $M$ is the maturity value of the bond.
Or, more compactly,
$$
P=\sum \frac{C}{(1+i)^{t}}+\frac{M}{(1+i)^{n}}
$$